DISJOINT PATHS IN PLANAR GRAPHS
A Thesis
Presented to
The Academic Faculty
by
Laura Sheppardson
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
Mathematics
Georgia Institute of Technology
June 2003
DISJOINT PATHS IN PLANAR GRAPHS
Approved by:
Professor Xingxing Yu, Advisor
Professor Thomas Morley
Professor Richard Duke
Professor Robert G. Parker, ISYE
Professor William T. Trotter
Date Approved
ACKNOWLEDGEMENTS
My thanks first to my advisor, Xingxing Yu. He has helped me find my strengths, and
directed me toward questions which truly intrigue me. He kept me moving forward in
moments of frustration, set high expectations, and provided support for me to meet them.
Thanks to Professors Richard Duke, Tom Trotter, Tom Morley, and Gary Parker for
their service on my thesis committee. I would also like to thank Robin Thomas and the
rest of the faculty at Georgia Tech for challenging me, for always being open to questions,
and for providing examples of people who are passionate about mathematics.
I benefited greatly by the influence of Professor Tom Storer in my undergraduate career
at the University of Michigan. His initial encouragement kept me in mathematics, and
played a role in bringing me back to it. I will strive to live up to his example as a mentor.
My fellow students have provided a wonderfully collaborative environment. I have
learned much from them, both academically and personally, and they have helped me
to maintain balance in my life. My thanks to them all. Those important to me are too
numerous to list, but include Jose-Miguel, Claudia, Jorge, Todd, Jacob, Bryan, Paul, Sam,
and Jay. Sean Curran has shared in the details of the degree process, and lightened my
load. My office-mates, in their turn, have provided invaluable moral support on a daily
basis. Victor Morales-Duarte helped me get through comprehensive exams, and reminded
me to step away from work occasionally. In addition to sharing the thesis experience, Sarah
Day has been my friend, my sounding board, and my most encouraging ally.
I am grateful to my friends outside Georgia Tech for their encouragement and understanding. Special thanks to Beth for her undying optimism, and to Gene and Chien, who
somehow managed to ask just the right questions.
Finally, thanks to my family. I am glad they raised me with a belief that I could do great
things. I am even more appreciative that they provided me the opportunities to discover
which things are truly great.
iii
TABLE OF CONTENTS
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
LIST OF FIGURES
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
I
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.1
Graph definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.2
Algorithmic complexity . . . . . . . . . . . . . . . . . . . . . . . . .
4
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2.1
Multicommodity flows . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2.2
General linkages . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.2.3
Graph minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.2
1.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
OBSTRUCTIONS AND REDUCTIONS . . . . . . . . . . . . . . . . . .
13
2.1
Terminology and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.2
Obstructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.3
Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.4
Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
III MAXIMIZING PATHS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
II
(2,3)-linkages
3.1
Defining a maximizing path . . . . . . . . . . . . . . . . . . . . . . . . . .
27
3.2
Examining path bridges
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
3.3
Location of X relative to a . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.4
Location of X relative to path . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.5
Separating X from a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
3.5.1
Defining common paths . . . . . . . . . . . . . . . . . . . . . . . . .
62
3.5.2
The case t0 = x1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
3.5.3
The case that T falls to the left of P . . . . . . . . . . . . . . . . .
71
iv
IV SPECIAL CASE |X| = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
V
92
4.1
Obstructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
4.2
Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
4.3
A special case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
4.4
Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
4.5
1-connected case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
PROOF OF GENERAL CASE . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.1
Independent inductive cases . . . . . . . . . . . . . . . . . . . . . . . . . .
100
5.2
Vertex of X on outer face
. . . . . . . . . . . . . . . . . . . . . . . . . . .
106
5.3
Remaining inductive cases . . . . . . . . . . . . . . . . . . . . . . . . . . .
117
5.4
Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
139
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
v
LIST OF FIGURES
1
Complete and complete bipartite examples
. . . . . . . . . . . . . . . . . .
2
2
Obstructions in O1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3
Example of 3-separation (G1 , G2 ) . . . . . . . . . . . . . . . . . . . . . . . .
14
4
Obstructions in O1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
5
Obstructions in O2 and O3 . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
6
Obstructions in O4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
7
Obstructions in O5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
8
Obstruction in O6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
9
Obstructions in O7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
10
Graphs satisfying property (R1) . . . . . . . . . . . . . . . . . . . . . . . . .
23
11
Graphs satisfying property (R2) . . . . . . . . . . . . . . . . . . . . . . . . .
25
12
The subgraph LP (shaded), where P is from a1 to x . . . . . . . . . . . . .
28
13
Bridges without LP attachment . . . . . . . . . . . . . . . . . . . . . . . . .
30
14
Where b1 , b2 ⊆ V (UP ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
15
Where X \ {x1 } is not contained in V (LP )
. . . . . . . . . . . . . . . . . .
34
16
Lemma 3.7, Claim 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
17
Lemma 3.7, Claim 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
18
Lemma 3.7, choice of path W . . . . . . . . . . . . . . . . . . . . . . . . . .
38
19
Lemma 3.7, Claim 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
20
Lemma 3.7, Claim 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
21
Lemma 3.7 final cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
22
Establishing separations when special paths exist . . . . . . . . . . . . . . .
44
23
x2 cofacial with a1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
24
Separations when b1 ∈ V (D) . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
25
Separations when s0 not cofacial with b2 . . . . . . . . . . . . . . . . . . . .
56
26
Combinations of separations, Case (5.1) . . . . . . . . . . . . . . . . . . . .
58
27
Case (5.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
28
Combinations of separations, Case (5.2) . . . . . . . . . . . . . . . . . . . .
61
29
Initial path definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
vi
30
Defining path R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
31
Where t0 = b2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
32
Where path Ox does not exist . . . . . . . . . . . . . . . . . . . . . . . . . .
67
33
Forcing crossed 3-separations when t0 = x1 . . . . . . . . . . . . . . . . . . .
68
34
Cases of crossed 3-separations . . . . . . . . . . . . . . . . . . . . . . . . . .
70
35
Where a2 is a cutvertex of LQ . . . . . . . . . . . . . . . . . . . . . . . . . .
71
36
Defining Z 0 (U ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
37
Where x1 not in LQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
38
Where x1 ∈ MQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
39
Defining separation (Hz , Mz ) . . . . . . . . . . . . . . . . . . . . . . . . . .
77
40
Example of z2 6= b1 , not cofacial s or s0 . . . . . . . . . . . . . . . . . . . . .
79
41
2-cut in LQ corresponding to 3-cut in LP
. . . . . . . . . . . . . . . . . . .
81
42
Where {w2 , s0 } ⊆ V (C). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
43
Where S0 and T0 both exist . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
44
Where s0 = a1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
45
x1 ∈ V (C),
(Γ, {x1 , x2 }) ∈ Θ4 . . . . . . . . . . . . . . . . . . . . . . . . .
95
46
Condition (4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
47
Subgraph definitions for (Γ, X 0 ) ∈ O5 . . . . . . . . . . . . . . . . . . . . . .
101
48
Case 1, x1 ∈ V (H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
102
49
(Γ, X 0 ) ∈ O6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
105
50
x1 ∈ V (C), X 0 ⊂ Int(D). . . . . . . . . . . . . . . . . . . . . . . . . . . . .
107
51
X1 = {x1 }, X2 = {x2 }, separation (Ok , Nk ). . . . . . . . . . . . . . . . . . .
111
52
X1 = {x1 }, X2 = {x2 }, leading to (Γ, X) ∈ O5 . . . . . . . . . . . . . . . . .
112
53
X1 = {x1 }, X2 = {x2 }, (Γ, X 0 ) ∈ O3 . . . . . . . . . . . . . . . . . . . . . . .
113
54
X2 = ∅, (Γ, X 0 ) ∈ O4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
116
55
Where X2 = ∅, (Γ, X 0 ) ∈ O7 . . . . . . . . . . . . . . . . . . . . . . . . . . .
117
56
Separation with X 0 * V (LP ) . . . . . . . . . . . . . . . . . . . . . . . . . .
118
57
Example of (Γ, X 0 ) ∈ O1 with X 0 * V (LP )
. . . . . . . . . . . . . . . . . .
120
58
Example of (Γ, X 0 ) ∈ O1 with X 0 ⊆ V (MP ) . . . . . . . . . . . . . . . . . .
122
59
(Γ, X 0 ) ∈ O1 , X 0 ⊆ V (MP ), (Γ, {x1 , x2 }) reducible . . . . . . . . . . . . . .
125
60
(Γ, X 0 ) ∈ O2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
129
vii
61
Where (Γ, X 0 ) ∈ O3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
132
62
Examples of (Γ, X 0 ) ∈ O7 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
135
63
Examples of (Γ, X 0 ) ∈ O7 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
137
64
(Γ, X 0 ) ∈ O7 with (Γ, X3 ) reducible . . . . . . . . . . . . . . . . . . . . . . .
138
65
(Γ, X 0 ) satisfies (R1), a01 = a1 . . . . . . . . . . . . . . . . . . . . . . . . . .
141
66
Disjoint paths in H when {a1 , a2 } ∩ V (K) = ∅ . . . . . . . . . . . . . . . .
143
67
When x1 ∈ V (Pi ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
144
68
When x1 ∈ V (Q1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
144
viii
SUMMARY
Disjoint paths in graphs, and disjoint subgraphs, have applications in such areas
as circuit design and multicommodity flows. They also lie at the heart of graph minor
questions. Here we consider a question of finding disjoint paths connecting specific subsets
of vertices in a planar graph. The specific form of that question is motivated by a problem
of Robertson and Seymour on (2, 3)-linkages.
Consider a plane graph G, an ordered set {a1 , b1 , a2 , b2 } of vertices lying on the outer
boundary of G, and a vertex set X disjoint from {a1 , b1 , a2 , b2 }. We ask when there exist
three disjoint paths: one between the vertices b1 and b2 , one from a1 to the set X, and one
from a2 to X. We establish a good characterization of those graphs which do not contain
such paths, called obstructions. The proof is constructive, implying an algorithm which
either produces the desired paths or identifies G as an obstruction.
The introductory chapter provides the basic definitions needed to understand the main
result. It also briefly discusses some applications giving rise to disjoint path questions. We
then describe a related graph structure question, regarding linkages, to which this result
contributes.
The remainder of the document is devoted to the details of the main result. We first
define obstructions, those graphs which do not have the desired paths. We then describe
graph reductions, which reduce the disjoint path question on a graph to the same question
on some smaller subgraph. Several technical tools and lemmas are then established. A
special case of the main result is proven, where the special set X includes just two vertices.
Finally, this special case is applied together with the technical lemmas to prove the general
case.
ix
CHAPTER I
INTRODUCTION
1.1
Background
This section provides the basic background in graph theory needed to understand our main
result. We define graphs, and paths in graphs. We briefly consider the ideas of connectivity
and cutsets as they relate to the existence of disjoint paths. We also review some basic
concepts of algorithmic complexity, which will be helpful when describing previous results
in the field.
1.1.1
Graph definitions
A graph consists of a set of vertices and a set of edges, where each edge is an unordered
pair of vertices. Given a graph G, we denote by V (G) and E(G) the vertex set and edge set
of G, respectively. All graphs considered here will have finite vertex and edge sets. Unless
otherwise specified, they will be simple, meaning that each pair of vertices appears at most
once in the edge set, and that the two vertices of each pair are distinct. Where u, v ∈ V (G),
we say u is adjacent to v if there is an edge uv ∈ E(G). We may also call v a neighbor of
u. The edge uv is incident with each of the vertices u and v.
We represent graphs visually by drawing them on some surface. The vertex set is
represented by points, and an edge is represented by a line connecting the two vertices with
which it is incident. We call a graph planar if it can be drawn in the plane in such a way
that no edges cross one another. A plane graph is one which is so drawn.
Two specific classes of graphs which will be mentioned in our discussion are complete
graphs and bipartite graphs. A graph is complete if every distinct pair of its vertices is
included in its edge set. We denote by Kn the complete graph on n vertices. A graph
G is bipartite if its vertices can be partitioned into two disjoint sets A and B such that
A ∪ B = V (G), and for any edge uv we have either u ∈ A, v ∈ B or u ∈ B, v ∈ A. A
1
(a) K5
(b) K3,3
Figure 1: Complete and complete bipartite examples
complete bipartite graph G is a bipartite graph with vertex partition A ∪ B = V (G) such
that for every vertex pair a ∈ A, b ∈ B, we have the edge ab ∈ E(G). The graph Km,n
is the complete bipartite graph with partition sets of size m and n respectively. Figure 1
shows the examples K5 and K3,3 .
A path in a graph can be thought of as a way of traveling from one vertex to another by
traversing edges of the graph. We require that no vertex be revisited in the course of this
traversal. Rigorously, a path is an ordered list of alternating vertices and edges, beginning
and ending with vertices, v0 , e1 , v1 , e2 , v2 , e3 , v3 , . . . , en , vn , where each edge ei is incident
with vi−1 and vi , and no vertex appears more than once in the list. Since we are dealing
with simple graphs only, a path can be determined completely by an ordered list of vertices.
Where v0 , v1 appear consecutively in the ordered list, we know the edge v0 v1 is included.
Where v0 and vn are the first and last vertices of the path, we say the path is from v0 to vn .
Note that since we are dealing with undirected graphs, a path from v0 to vn is associated
with a path from vn to v0 in the obvious way. In most cases, we will make no distinction
between the two.
Paths may also be considered as special cases of subgraphs. We say a graph H is a
subgraph of G, denoted H ⊆ G, if V (H) ⊆ V (G) and E(H) ⊆ E(G). Suppose we have two
different subgraphs, H1 , H2 ⊆ G. We call these disjoint if they have no vertices in common.
That is, V (H1 ) ∩ V (H2 ) = ∅. This definition of disjoint is sometimes called vertex-disjoint.
One could also consider edge-disjoint subgraphs, requiring E(H1 ) ∩ E(H2 ) = ∅. Notice that
if two subgraphs are vertex-disjoint, they are necessarily edge-disjoint.
2
Thinking of a path as a way “to get there from here” in a graph, it is natural to ask
when a path exists between two vertices. We call a graph G connected if for each pair of
vertices u, v ∈ V (G), there is some path from u to v in G. We may further ask whether
there are multiple paths from u to v. In order to make sense of this question, we need to
be clear about what we mean by multiple paths. Suppose we have many paths from u to v,
but every one of them passes through vertex x. Then removal of vertex x destroys all paths
from u to v, and G − x is disconnected. (Here G − x is the graph formed from G by removal
of vertex x, and all edges incident with x.) We might think of paths as subgraphs, and say
that two paths are independent if they are disjoint. If we want their end vertices to be the
same, however, we need a modification. Given a path P from u to v, let us call all vertices
in V (P ) except u and v internal vertices of P . Two paths P and Q will be called internally
disjoint if they share no internal vertices. (Some authors call this openly disjoint.)
We now extend the definition of connectivity, and consider its implications for disjoint
paths. A cutset of G is a set of vertices T ⊆ V whose removal produces a disconnected
graph. If T is a cutset with |T | = k, it is called a k-cut in G. A graph G is called n-connected
if |V (G)| ≥ k + 1, and G has no k-cut with k < n. A classic result of Menger shows that G
is k-connected if and only if it contains k internally disjoint paths between any two vertices.
(This formulation is due to Whitney (1932).)
Menger’s theorem actually tells us more than this. Given vertex sets A, B ⊆ V (G), we
call any path from a ∈ A to b ∈ B an A-B path. If T ⊆ V is such that every A-B path
includes some vertex of T , then T is said to separate A from B. Menger’s theorem tells us
that the minimum number of vertices separating A from B in G is equal to the maximum
number of disjoint paths from A to B.
The existence of multiple disjoint paths in a graph is certainly intimately related to
high connectivity. If we ask for paths in some specific configuration, however, the exact
relationship is not obvious. Suppose, for example, we have disjoint set A, B ⊆ V (G), each
of size k. If no set of less than k vertices separates A from B in G, then there are k
disjoint paths from A to B in G. This gives us no control, however, over which vertices are
connected. Do we have a path from a1 ∈ A to b1 ∈ B? Or can we get k disjoint paths
3
only if we include a path from a1 to b2 ? The question of finding paths connecting specific
subsets or pairs of vertices is what we will consider here.
1.1.2
Algorithmic complexity
When considering applications, we will be concerned with not only knowing whether disjoint
paths exist, but actually finding them. Where algorithms are available to do this, we will
want some way to measure their efficiency. To this end, we consider several ideas from
complexity theory. A complete treatment of the definitions presented here can be found in
[2].
The running time of an algorithm is measured by counting the number of operations
performed. This assumes that we have a consistent definition of “operation”. We generally
consider worst-case running times. We use the notation f (n) = O(g(n)) to mean that the
asymptotic growth of f (n) is bounded above by that of g(n). That is, there exist positive
constants c and n0 such that 0 ≤ f (n) ≤ cg(n) for all n ≥ n0 . We say an algorithm is
polynomial time if the worst case running time is O(nr ) for some fixed constant r, where n
is the size of the input.
We say a problem is polynomially solvable if there is some polynomial time algorithm
which will solve any instance of that problem. The complexity class P is the set of all
problems which are polynomially solvable.
The complexity class N P (nondeterministic polynomial time) is a bit more difficult to
describe concisely. This applies to problems for which a proposed solution can be verified in
polynomial time, but no algorithm is known to find a solution in polynomial time. Suppose
for example that we are given vertex lists in a graph, with the claim that they represent
disjoint paths in that graph. We can verify the claim quite easily. This is distinct from
being able to find sets of disjoint paths when given a graph.
An NP-complete problem is in some sense a representative of the class N P . The complexity class of a problem is often determined by a technique called reduction. If we can
reduce problem Y to problem X, then any algorithm which will solve X can be modified
to solve problem Y . A problem X is N P -complete if every other problem in N P can be
4
reduced to X. If we were to find a polynomial time algorithm to solve any N P -complete
problem, we would be able to solve all N P -complete problems in polynomial time. This
would show P = N P , a conclusion there is extremely strong evidence against.
In most cases, we show a problem is in P by exhibiting a polynomial time algorithm
to solve it. However, results in the field of graph minors (specifically, the excluded minor
results of Robertson and Seymour [13]) sometimes allow us to state that a polynomial time
algorithm does exist, without specifying what that algorithm may be.
1.2
Applications
We now consider some of the applications of disjoint paths and disjoint subgraphs. Our
discussion of physical applications is limited to multicommodity network flows, but this
is certainly not the only example. Disjoint paths are highly studied in the field of VLSI
(very large systems integration), for instance, as they have clear application in the design of
circuit layouts. Most physical examples of disjoint path problems, however, are conceptually
similar to the flows we describe.
Our main result, somewhat technical in nature, has direct applications within graph
theory. We discuss these in the form of linkages and graph minors.
1.2.1
Multicommodity flows
Network flows are a well-studied topic in the field of optimization, and have become a fairly
standard example of a physical application of graph theory concepts. Here a graph is used to
model a physical network, where something is being transported from node to node (vertex
to vertex), along pathways defined by the graph’s edges. This may be used to model such
systems as electrical networks, telecommunications, or the transportation of physical goods,
whether within a single production plant or across a national distribution system.
Networks are generally modeled by directed graphs. In a directed graph, each edge is
an ordered pair of vertices. An edge (u, v) is thought of as an edge from u to v. A network
is a directed, connected graph G = (V, E) with two special nodes s and t, which we will call
the source and the sink respectively. Each edge is assigned a non-negative capacity, c(u, v).
5
The source node generally has only outgoing edges, and the sink only incoming edges. A
flow on a network is a function f : E → R+ satisfying two sets of conditions:
1. For each (u, v) ∈ E, f (u, v) ≤ c(u, v).
2. For each fixed v ∈ V \ {s, t},
P
(u,v)∈E
f (u, v) =
P
(v,w)∈E
f (v, w).
The first condition simply requires that the flow on each edge not exceed the capacity
of that edge. The second ensures conservation of flow at each node. If we think of the flow
as representing the volume of material moving along a transportation network, we may see
material as being produced at the source, consumed at the sink, and moving without loss
P
through all internal nodes. The value of a flow is simply (s,v)∈E f (s, v), or equivalently
P
(v,t)∈E f (v, t), the net quantity being transported from source to sink.
The usual question asked in the context of network flows is, given a particular network,
what is the flow of maximum value? Or, generally more useful in practice, what is the
maximum integer-valued flow? (In distributing a product, for example, where we have
a some predetermined shipping unit.) When asking for integer-valued flows, one generally
restricts capacities to integer values as well. There are now fairly standard algorithms which
can determine the maximum flow on a network in O(|V |3 ) time.
A multicommodity flow extends the standard network flow formulation to a network
with multiple sources and sinks. We again have a directed graph G = (V, E), now with
{s1 , . . . , sk , t1 , . . . tk } ⊆ V , constituting a source si and sink ti for each of k commodities.
The edges of E are considered capacity edges. A demand edge (ti , si ) is added between
each sink/source pair, with a demand (or request) value r(si , ti ). Call the set of demand
edges D. In principal, we might think of a flow as a multivalued function F : E ∪ D → Rk ,
where each coordinate Fi represents the flow of commodity i, again subject to capacity and
conservation conditions. The technical formulation used is actually a bit different, as we
now describe.
We wish to find a collection C of circuits in G, and a function F : C → R+ , satisfying
the following conditions:
1. Each C ∈ C contains exactly one demand edge.
6
2. For each d ∈ D,
3. For each e ∈ E,
P
d∈C∈C
F (C) = r(d).
e∈C∈C
F (C) ≤ c(e).
P
We can think of each circuit as representing the flow of a single commodity. This is established by the first condition above. The second condition ensures that the demand for each
commodity is met, and the third that no edge capacities are exceeded.
If we consider an integer-valued multicommodity flow problem where each edge capacity
and each demand is 1, we have the edge-disjoint path problem. Since each edge can carry
only one commodity, any flow will specify a collection of k paths, each from si to ti , where
no pair of paths has any edge in common. Conversely, any collection of such paths defines
a flow. If we further require that at most one commodity may pass through each node, we
have the vertex-disjoint path problem. This is equivalent to the k-linkage problem, which
will be discussed in Section 1.2.2.
Sebo [16] gives a polynomial time algorithm which determines the integer solvability (i.e.,
whether a flow exists) when the graph representing capacity and demand is planar, and the
number of commodities is bounded by a predetermined integer. While polynomial time
algorithms are available under some restrictions, no general characterization for solvability
of multicommodity flows is known.
1.2.2
General linkages
Given a graph G, consider an ordered set of distinct vertices X = {s1 , . . . , sk , t1 , . . . tk } ⊆
V (G), which we will call terminals. A collection of disjoint paths P1 , . . . , Pk in G, with each
Pi from si to ti , is called an X-linkage, or a k-linkage. Suppose G is a graph with at least
2k vertices, and for every set X of 2k terminals, G has an X-linkage. Then G is called
k-linked.
Here we recall our earlier discussion on graph connectivity. Suppose we are given a
graph G and sets S = {s1 , . . . , sk }, T = {t1 , . . . tk } ⊆ V (G). If G is k-connected, then
by Menger’s theorem we have k disjoint paths from S to T . This does not ensure that
there is an X-linkage. High enough connectivity, though, does ensure that a graph is klinked. Results by Jung [6] and Larman [8], for example, show that 2k-connected graphs
7
with certain subgraphs are indeed k-linked. A result of Mader [10] shows that the necessary
subgraph can be guaranteed by some minimal edge density. High edge density can in turn
be guaranteed by high connectivity. So there is some function f (k) : N → N such that every
f (k)-connected graph is k-linked. Bollobás and Thomason give the first linear bound on
this f (k) in [1]. They show that any graph which is 22k-connected is k-linked.
Can we easily determine, given a specific instance of a graph G, whether G is k-linked?
Karp showed in 1975 [7] that this question is NP-complete. Lynch [9] showed that the
question remains NP-complete even if we restrict our input to graphs which are planar.
Middendorf and Pfeiffer [11] extended this to a further restricted class, where the graph
formed from G by the addition of an edge between si and ti for each terminal pair is planar.
These results all consider k as part of the input.
When k is fixed, the algorithmic outlook is more hopeful. In [14], Robertson and Seymour describe an algorithm which for fixed k has running time O(|V (G)|3 ), and determines
whether a graph G has an X-linkage for specified X = {s1 , . . . , sk , t1 , . . . tk } ⊆ V (G). Unfortunately, this algorithm is “not practically feasible, since it involves the manipulation of
enormous constants”. In [12], Reed et al. show that there is a linear time algorithm answering this question for planar graphs, where again k is fixed. This result can be extended
to any fixed compact surface.
The linear time algorithm of Reed et al. actually addresses a more general question.
It determines whether, given vertex sets X1 , X2 , . . . Xk ⊂ V (G), there exist disjoint trees
T1 , . . . Tk in G, with each Xi ⊆ V (Ti ). A tree here is a connected graph for which each vertex
is a cutvertex. (An equivalent definition of a tree is a connected graph which contains no
cycles.) Disjoint paths are special cases of disjoint trees, where |Xi | = 2 for each i = 1, . . . , k.
Polynomial time algorithms do not extend to directed graphs, where the question of the
existence of a k-linkage remains NP-complete even when k is fixed at 2. In the remainder
of our discussions, then, we will restrict ourselves to undirected graphs. In the case of
multicommodity flows, this corresponds to an assumption that edges appear in pairs (u, v)
and (v, u), so that there may be flow in either direction between two adjacent vertices.
8
We have seen that there are some known sufficient conditions for a graph to be klinked. No full characterization of k-linked graphs is known. We have also seen that there
are algorithms for determining whether specific linkages exist in a given graph. These
algorithms generally construct the desired linkage, or show that it does not exist. The
lemma below (Lemma 1.1) demonstrates the type of characterization we might hope for, in
this case for 2-linkages. This result, now well-known, is proven by Thomassen in [18]. It
was also obtained (but stated without proof) by Seymour [17]. The conditions described in
(2) can be thought of as planar embedding “modulo 3-cuts”.
Lemma 1.1. Given a 2-connected graph G and vertices s1 , s2 , t1 , t2 ∈ V (G), then either
1. there are disjoint paths from s1 to t1 and from s2 to t2 , or
2. for some n ≥ 0 there are pairwise disjoint sets A1 , . . . , An ⊆ V (G) \ {s1 , s2 , t1 , t2 }
such that
(a) for i 6= j, no vertex of Ai has a neighbor in Aj ,
(b) each Ai has at most three neighbors in V (G) \ Ai , and
(c) where G0 is the graph formed from G by, for each 1 ≤ i ≤ n, removing the vertices
of Ai and adding edges between each pair of neighbors of Ai , G0 can be embedded
in a disk with s1 , s2 , t1 , t2 appearing in order along the boundary of the disk.
Here we are not concerned with exhibiting a specific linkage. We do have a very simple
description of the structure which obstructs the existence of a linkage. This has proven
useful in examining other graph structure properties. Our approach to (2, 3)-linkages, for
example, will rely on this lemma as a basic tool.
1.2.3
Graph minors
Given a graph G, a graph H is called a minor of G if it can be obtained from a subgraph
of G by contracting edges. Contraction of an edge uv ∈ E(G) is performed by replacing the
two vertices u and v with a single new vertex, adjacent to each neighbor of u or v.
Perhaps the most widely known result on graph minors is Kuratowski’s Theorem: A
graph is planar if, and only if, it does not contain K5 or K3,3 (Figure 1) as a minor. (The
9
statement in terms of minors is actually by Wagner (1937). Kuratowski’s version (1930)
is in terms of subdivisions.) Graph minors in general are closely related to questions of
surface embeddings. For example, within their extensive series of papers on graph minors
[13], Robertson and Seymour have shown that for any surface, there is a finite collection of
graphs such that every graph not embeddable in the surface contains one member of this
collection as a minor. Graph minors also have applications to graph coloring and to many
other structural graph properties.
Where H is a minor of the graph G, vertices of H are realized as disjoint connected
subgraphs of G, and edges of H as internally disjoint paths between those subgraphs. So
when we ask questions about graph minors, we are really asking about the existence of
particular disjoint subgraphs and paths.
1.3
(2,3)-linkages
The specific disjoint path problem considered in this thesis is motivated by the work of
Robertson, Seymour, and Thomas addressing Hadwiger’s conjecture. The conjecture, made
in 1943 in [5], concerns graph minors and colorings. A graph is said to be t-colorable if
one can color its vertices in such a way that no two vertices of the same color are adjacent,
using at most t colors.
Hadwiger Conjecture. For every integer t ≥ 0, every loopless graph with no Kt+1 minor
is t-colorable.
This is a generalization of the famous four-color theorem, stating that all planar graphs
are 4-colorable, since no planar graph contains a K5 minor. The Hadwiger conjecture is
known to be true for t ≤ 3, and is equivalent to the four-color theorem for t = 4. In
[15], Robertson et al. show that every minimal counterexample to the Hadwiger conjecture
for t = 5 is an apex graph. That is, removal of a single vertex (the apex) leaves a planar
graph. Hence the Hadwiger conjecture for t = 5 is also equivalent to the four-color theorem.
The details of their proof, carefully examining the structure any minimal counterexample
must have, led them to consider the following question: Given a graph G and vertices
a0 , a1 , a2 , b1 , b2 ∈ V (G), when does G have disjoint connected subgraphs A and B with
10
a1
a0
1
0
0
1
1
0
0
1
0
1
a1
1
0
0
1
0
1
11
00
00
11
00
11
b1
b2
b2
11
00
11
00
1
0
0
1
a2
1
0
0
1
b1
0
1
1
0
0
1
00
11
11
00
00
11
11
00
00
11
a0
a2
Figure 2: Obstructions in O1
{a0 , a1 , a2 } ⊆ V (A) and {b1 , b2 } ⊆ V (B)? We will call such a configuration, if it exists, a
(2, 3)-linkage (see Figure 2(a)).
The results of this thesis can be implemented in characterizing when a (2, 3)-linkage
exists. We briefly outline an approach to that proof here. Given G and a0 , a1 , a2 , b1 , b2 ∈
V (G), consider the graph G0 = G − a0 . We ask first whether G0 has disjoint paths P from
a1 to a2 and Q from b1 to b2 . If these paths do exist, contract path P in G to a single
vertex, forming the graph G̃. Then ask for disjoint paths from a0 to P and from b1 to b2
in G̃. In both of these steps, we are asking for 2-linkages. From Lemma 1.1 we have a
complete characterization of those graphs which do not have the desired 2-linkages. This
can be extended to a characterization of those graphs which do not have a (2, 3)-linkage.
If the paths P and Q do not exist in G0 , then by Lemma 1.1, G0 can be embedded in
the plane, modulo 3-cuts, with a1 , b1 , a2 , b2 appearing in order on its outer cycle. If we have
a plane embedding of G0 then, with the addition of the vertex a0 , we see that G is an apex
graph (see Figure 2(b)). Let X be the set of neighbors of a0 . We want disjoint paths in G0 :
from a1 to X, from a2 to X, and from b1 to b2 . So if we have a characterization of when
these paths exist in G0 , we can determine when a (2, 3)-linkage exists in G.
Notice that if the graph G is 3-connected, the graph G0 is 2-connected. Suppose there
is a 2-cut Z in G0 . Each component of G0 − Z must contain some vertex of X. Keeping in
mind that we want a path from ai to X, we see that if a component of G0 − Z includes none
of a1 , b1 , a2 , b2 , we need not consider the details of the graph structure in that component. If
11
we can find a path from ai to one of the vertices of Z, then we can find a path from ai to any
vertex of X contained in that component. We know from Lemma 1.1 that we will not make
use of more than one vertex of X in that component, since this would not allow for a path
from b1 to b2 . These observations will motivate our use of (3, {a1 , b1 , a2 , b2 })-connectivity
in the definition of disk graphs, Definition 2.2.
12
CHAPTER II
OBSTRUCTIONS AND REDUCTIONS
2.1
Terminology and Notation
All graphs considered here will be finite and simple. Where there is no confusion, we
may use x ∈ G to mean x ∈ V (G) or x ∈ E(G) for any graph G. Given two graphs
G1 and G2 , we define G := G1 ∪ G2 to be the graph with V (G) = V (G1 ) ∪ V (G2 ) and
E(G) = E(G1 ) ∪ E(G2 ).
When G is a connected graph which is not 2-connected, we can describe the structure
of G using Tutte’s block graph construction [19]. A cut edge is any edge e ∈ E(G) whose
removal disconnects G. A block of G is a maximal 2-connected subgraph, or a subgraph
induced by a cut edge. We construct a graph Blk(G) such that V (Blk(G)) = {cutvertices
ui of G} ∪ {blocks Bj of G}, and ui is adjacent to Bj in Blk(G) if ui ∈ V (Bj ) in G. We
call a block B of G an endblock of G if B contains at most one cutvertex of G. If the graph
Blk(G) is a simple path, we say that G is a chain of blocks. Note that for a connected graph
G this is equivalent to the condition that each block of G contains at most two cutvertices
of G, and each cutvertex of G is contained in at most two blocks of G.
Let P be a path in a graph G, and let u, v ∈ V (P ); then P [u, v] denotes the subpath of P
between u and v, inclusive. P (u, v) denotes the same subpath with its endpoints excluded,
and P (u, v], P [u, v) denote the subpath with the obvious endpoint excluded. Furthermore,
if G is a plane graph and C is a cycle in G, then for any distinct u, v ∈ V (C), we use C[u, v]
to indicate the subpath of C from u to v, inclusive, which follows the clockwise orientation
of C. Define C(u, v), C(u, v], C[u, v) in the obvious way.
Given vertex sets X, Y in a graph G, we say a path P is a path from X to Y in G
if P has endpoints x ∈ X and y ∈ Y , and is internally disjoint from X ∪ Y . That is,
V (P ) ∩ X = {x} and V (P ) ∩ Y = {y}.
Given any plane graph G, we let ∂G be the subgraph of G consisting of vertices and
13
v11
0
1
0
0
1
11
00
11
00
G2
v2
G1
00
00
11
v3 11
1
0
0
1
v4
Figure 3: Example of 3-separation (G1 , G2 )
edges incident with the infinite face of G. When G is 2-connected, ∂G is a cycle, and we
call this the outer cycle of G.
We say a collection of vertices in a plane graph G are cofacial if they are incident with
a common face of G.
Recall that if G is a connected graph, an k-cut of G is a set of vertices T ⊆ V (G) with
|T | = k such that G − T has at least two components. G is called n-connected if it has no
k-cut with k < n. We now define something similar to a k-cut, called a k-separation.
Definition 2.1. Let G1 and G2 be edge-disjoint subgraphs of a graph G. We say (G1 , G2 )
is a separation of G if V (G) = V (G1 ) ∪ V (G2 ), E(G) = E(G1 ) ∪ E(G2 ), E(G1 ) 6= ∅, and
E(G2 ) 6= ∅. If |V (G1 ∩ G2 )| = k, we say (G1 , G2 ) is a k-separation of G, or a separation of
order k.
Figure 3 illustrates a 3-separation (G1 , G2 ) with V (G1 ∩ G2 ) = {v1 , v2 , v3 }. Notice that
the illustrated graph also has a 4-separation (G01 , G02 ) with V (G01 ∩ G02 ) = {v1 , v2 , v3 , v4 }.
In the case of separation (G1 , G2 ), we can find a simple closed curve Ω in the plane whose
intersection with G is exactly V (G1 ∩ G2 ), and which separates V (G1 )\V (G2 ) from V (G2 )\
V (G1 ) in the disk in which G is embedded. In the case of separation (G01 , G02 ), no such closed
curve exists. In several later definitions, we will desire the existence of such a closed curve.
We call the separation (G1 , G2 ) non-degenerate if |V (G1 ∩ G2 )| = k, and V (G1 ∩ G2 ) can be
labeled as {g1 , g2 , . . . , gk } in such a way that gi is cofacial with gi+1 for each i ∈ {1, . . . , k−1},
and gk is cofacial with g1 .
14
Definition 2.2. Let G be a graph and let a1 , b1 , a2 , b2 be distinct vertices of G. If for
any T ⊆ V (G) with |T | ≤ 2, each component of G − T contains at least one element of
{a1 , b1 , a2 , b2 }, then G is said to be (3, {a1 , b1 , a2 , b2 })-connected.
We call Γ = (G, a1 , b1 , a2 , b2 ) a disk graph if G is a 2-connected graph drawn in a disk
in the plane with no pair of edges crossing such that a1 , b1 , a2 , b2 appear in clockwise order
along the boundary of the disk, and G is (3, {a1 , b1 , a2 , b2 })-connected.
Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph, and let X ⊆ V (G) \ {a1 , b1 , a2 , b2 }. We ask
whether three disjoint paths exist in G: from a1 to X, from a2 to X, and from b1 to b2
respectively. If these paths exist, we say X is feasible in Γ. Otherwise, X is infeasible in Γ.
We will show that if X is infeasible in Γ, then either Γ can be “reduced” (to be described
in Section 2.3), or Γ falls into one of several families of graphs (to be described in Section
2.2).
2.2
Obstructions
Here we describe seven classes of disk graphs Γ = (G, a1 , b1 , a2 , b2 ) and vertex sets X ⊆
V (G) \ {a1 , b1 , a2 , b2 } for which X is infeasible in Γ. In all figures, shaded areas represent
possible locations for vertices of X.
Definition 2.3. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C, and let
X ⊆ V (G) \ {a1 , b1 , a2 , b2 } with |X| ≥ 2. We say (Γ, X) ∈ O1 if all x ∈ X are incident with
a common finite face, or X ⊆ V (C[b1 , b2 ]), or X ⊆ V (C[b2 , b1 ]). See Figure 4.
Proposition 2.1. If (Γ, X) ∈ O1 , then X is infeasible in Γ.
Proof. Suppose there are disjoint paths P1 from a1 to x1 ∈ X and P2 from a2 to x2 ∈ X.
Since x1 and x2 are cofacial, V (P1 ) ∪ V (P2 ) separates b1 from b2 in G. Since this is true for
any choice of P1 and P2 , X is infeasible in Γ.
Definition 2.4. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph, and let X ⊆ V (G)\{a1 , b1 , a2 , b2 }
with |X| ≥ 2. We say (Γ, X) ∈ O2 if G has a separation (H, K) of order 3 such that
X ⊆ V (K), {a1 , a2 , b1 , b2 } ⊆ V (H). See Figure 5(a).
15
ai
a1
X
b1
b2
b3−i
bi
a3−i
a2
Figure 4: Obstructions in O1
ai
a1
K
b2
b1
H
b3−i
bi
a3−i
a2
(a) Class O2
(b) Class O3
Figure 5: Obstructions in O2 and O3
Proposition 2.2. If (Γ, X) ∈ O2 , then X is infeasible in Γ.
Proof. Label V (H ∩ K) = {h1 , h2 , h3 }. Suppose there are disjoint paths P1 from a1 to X
and P2 from a2 to X. Then Pi ∩ V (H ∩ K) 6= ∅ for i ∈ {1, 2}. Assume by symmetry that
h1 ∈ V (P1 ) and h2 ∈ V (P2 ). Now h1 and h2 are cofacial, so V (P1 ) ∪ V (P2 ) separates b1
from b2 in G. Since this is true for any choice of P1 and P2 , X is infeasible in Γ.
Definition 2.5. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C, and let
X ⊆ V (G) \ {a1 , b1 , a2 , b2 } with |X| ≥ 2. We say (Γ, X) ∈ O3 (Figure 5(b)) if there is some
i ∈ {1, 2} such that for each x ∈ X, one of the following holds:
1. x ∈ C[b3−i , bi ], or
16
2. there is a finite face of G incident with both ai and x, or
3. G has a separation (H, K) of order 3 such that ai ∈ V (H∩K), x ∈ V (K), {a1 , a2 , b1 , b2 } ⊆
V (H)
Proposition 2.3. If (Γ, X) ∈ O3 , then X is infeasible in Γ.
Proof. Assume by symmetry that the definition above is satisfied with i = 1. Let P2 be any
path from a2 to X. Then some vertex of P2 is cofacial with a1 , and V (P2 ) ∪ {a1 } separates
b1 from b2 in G. Hence X is infeasible in Γ.
For the following definition, refer to Figure 6
Definition 2.6. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C, and let
X ⊆ V (G) \ {a1 , b1 , a2 , b2 } with |X| ≥ 2. Suppose there is some i ∈ {1, 2} such that
X ∩ C[b3−i , bi ] 6= ∅. Partition X into X1 , X2 , X3 with X1 = X ∩ V (C[b3−i , ai ]), X2 =
X ∩ V (C[ai , bi ]), X3 = X \ (X1 ∪ X2 ). We say (Γ, X) ∈ O4 if there is some j ∈ {1, 2},
x0 ∈ Xj , and Ca = C[ai , x0 ] if x0 ∈ C[ai , bi ], Ca = C[x0 , ai ] if x0 ∈ C[b3−i , a1 ], such that
Xj ∩ V (Ca ) = {x0 } and one of the following holds:
1. X3−j = ∅, and
(a) there is a finite face of G incident with some vertex in V (Ca ) and all vertices in
X3 (see Figure 6(a)), or
(b) G has a separation (H, K) of order 3 with Xj ∪ {a1 , b1 , a2 , b2 } ⊆ V (H), X3 ⊆
V (K), and |V (H ∩ K) \ V (Ca )| ≤ 2 (Figure 6(b) and 6(c)), or
(c) |X3 | = 1, and G has a non-degenerate separation (H, K) of order 4 with Xj ∪
{a1 , b1 , a2 , b2 } ⊆ V (H), X3 ⊆ V (K), and |V (H ∩ K) \ V (Ca )| ≤ 2 (Figure 6(d)),
or
2. X3−j 6= ∅, and
(a) G has a 2-separation (H, K) with X1 ∪ X2 ∪ {a3−i , b1 , b2 } ⊆ V (H), X3 ∪ {ai } ⊆
V (K), (Figure 6(e)), or
17
(b) |X3 | = 1, and G has a non-degenerate separation (H, K) of order 4 with X1 ∪
X2 ∪ {a3−i , b1 , b2 } ⊆ V (H), X3 ∪ {ai } ⊆ V (K) (Figure 6(f)).
Proposition 2.4. If (Γ, X) ∈ O4 , then X is infeasible in Γ.
Proof. Assume by symmetry that the definition above is satisfied with i = 1, and Ca =
C[b2 , a1 ]. Let P be a path from a2 to x ∈ X. If V (P ) ∩ V (C[b2 , b1 ]) 6= ∅, then V (P )
separates b1 from b2 in G. We may assume, then, that V (P ) ∩ V (C[b2 , b1 ]) = ∅, and x ∈ X3 .
Suppose X2 = ∅ and there is a finite face of G incident with some vertex in V (Ca )
and all vertices in X3 . Then there is a vertex v ∈ V (Ca ) cofacial with x, and V (P ) ∪ {v}
separates X1 ∪ {b2 } from {a1 , b1 } in G. Since this is true for any choice of path P , X is
infeasible in Γ.
Now suppose G has a separation (H, K) of order at most 4 with {a1 , b1 , a2 , b2 } ⊆ V (H),
x0 ⊆ V (K), and |V (H ∩ K) − V (Ca )| ≤ 2. Now V (P ) ∩ V (H ∩ K) 6= ∅, so we have some
vertex vinV (Ca ) cofacial with some vertex p ∈ V (P ), and V (P ) ∪ {v} separates X1 ∪ {b2 }
from {a1 , b1 }. If X2 6= ∅, we find that p is also cofacial with some vertex v 0 ∈ C[a1 , b1 ].
Hence V (P ) ∪ {v 0 } separates X2 ∪ {b1 } from {a1 , b2 }. Since this is true for any choice of
path P , X is infeasible in Γ.
The following definition is illustrated in Figure 7.
Definition 2.7. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C, and let
X ⊆ V (G) \ {a1 , b1 , a2 , b2 } with |X| ≥ 2. We say (Γ, X) ∈ O5 if G has separations
(H1 , M1 ), (H2 , M2 ), and (H3 , M3 ), and a cycle D bounding a finite face of G such that the
following hold for some i ∈ {1, 2}:
1. V (H1 ∩ M1 ) = {u1 , v1 , w1 }, w1 ∈ V (C), {u1 , v1 } ⊆ V (D), {a1 , a2 , b1 , b2 } ⊆ V (H1 ),
2. V (H2 ∩ M2 ) = {u2 , v2 , w2 }, w2 ∈ V (C), {u2 , v2 } ⊆ V (D), {a1 , a2 , b1 , b2 } ⊆ V (H2 ),
3. V (H3 ∩ M3 ) = {u1 , u2 } ∪ ({w1 , w2 } ∩ {b1 , b2 }), ai ∈ V (M3 ), {a3−i , b1 , b2 } ∈ V (H3 ),
4. if w1 ∈
/ {b1 , b2 }, then u1 ∈ V (C), and if w2 ∈
/ {b1 , b2 }, then u2 ∈ V (C), and
18
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(f)
Figure 6: Obstructions in O4
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a3−i
(c)
Figure 7: Obstructions in O5
5. X ⊆ V (M1 ∪ M2 ∪ D ∪ C[b3−i , w1 ] ∪ C[w2 , bi ]),
Proposition 2.5. If (Γ, X) ∈ O5 , then X is infeasible in Γ.
Proof. Assume by symmetry that the above definition is satisfied for i = 1, so a1 ∈ V (M3 ).
Suppose there are disjoint paths P1 from a1 to X and P2 from a2 to X, each disjoint from
{b1 , b2 }. Note that V (P1 ) ∩ V (D) 6= ∅. If V (P2 ) ∩ V (C[b2 , w1 ] ∪ C[w2 , b1 ]) 6= ∅, then V (P2 )
separates b1 from b2 in G. Hence we may assume V (P2 ) ∩ V (C[b2 , w1 ] ∪ C[w2 , b1 ]) = ∅, so
V (P2 ) ∩ V (D) 6= ∅. But then V (P1 ) ∪ V (P2 ) separates b1 from b2 in G, and X is infeasible
in Γ.
Definition 2.8. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C, and let
20
ai
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w1
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1
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a3−i
Figure 8: Obstruction in O6
X ⊆ V (G) \ {a1 , b1 , a2 , b2 } with |X| ≥ 2. We say (Γ, X) ∈ O6 if G has separations (H1 , M1 )
and (H2 , M2 ) and a cycle D bounding a finite face of G such that the following hold for
some i ∈ {1, 2} (Figure 8):
1. V (D ∩ C) = {u1 , u2 }
2. For each j ∈ {1, 2} there is a path Cj ⊆ C from uj to {b1 , b2 } with V (Cj )∩{a1 , a2 } = ∅
3. V (Hj ∩ Mj ) = {uj , vj , wj }, vj ∈ V (D), wj ∈ V (Cj ), {a1 , b1 , a2 , b2 } ⊆ V (Hj ), and
4. X ⊆ V (M1 ∪ M2 ∪ D ∪ C1 ∪ C2 )
Proposition 2.6. If (Γ, X) ∈ O6 , then X is infeasible in Γ.
Proof. Assume by symmetry that a1 , u2 , w2 , b1 appear in clockwise order on C, as in Figure
8. Suppose there are disjoint paths P1 from a1 to X and P2 from a2 to X, each disjoint from
{b1 , b2 }. If V (P1 ) ∩ V (C[w1 , b2 ]) 6= ∅, we have V (P1 ) separating b1 from b2 in G. Similarly,
if V (P2 ) ∩ V (C[w2 , b1 ]) 6= ∅, then V (P2 ) separates b1 from b2 in G. So by planarity we may
assume V (P1 ) ∩ V (D) 6= ∅ and V (P2 ) ∩ V (D) 6= ∅, and hence V (P1 ) ∩ V (P2 ) separates b1
from b2 in G, and X is infeasible in Γ.
Definition 2.9. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C, and let
X ⊆ V (G)\{a1 , b1 , a2 , b2 } with |X| ≥ 2. We say (Γ, X) ∈ O7 if G has a 3-separation (H, K)
and a cycle D bounding a finite face of G such that for some i ∈ {1, 2}, the following hold
1. V (H ∩ K) = {u, v, w}, w ∈ C(b3−i , ai ) ∪ C(ai , bi ), and u, v ∈ V (D)
21
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a3−i
a3−i
(a)
(b)
Figure 9: Obstructions in O7
2. {a1 , a2 , b1 , b2 } ⊆ V (H), X ∩ V (K) * V (H),
3. if w ∈ C(b3−i , ai ), then u ∈ C(w, ai ] and X ⊆ V (K ∪ D ∪ C(b3−i , w]) (Figure 9(a)), if
w ∈ C(ai , bi ), then u ∈ C[ai , w) and X ⊆ V (K ∪ D ∪ C[w, bi )), and
4. if w ∈ {b3−i , bi }, then X ⊆ V (K ∪ D) (Figure 9(b)).
Proposition 2.7. If (Γ, X) ∈ O7 , then X is infeasible in Γ.
Proof. Assume by symmetry that w ∈ C[b2 , a1 ]. Suppose there are disjoint paths P1 from
a1 to X and P2 from a2 to X, each disjoint from {b1 , b2 }. If V (P2 ) ∩ V (C[b2 , w]) 6= ∅,
then V (P2 ) separates b1 from b2 in G. Hence we may assume V (P2 ) ∩ V (C[b2 , w]) = ∅, so
V (P2 ) ∩ V (D) 6= ∅. By planarity, we find V (P1 ) ∩ V (D) 6= ∅, so V (P1 ) ∪ V (P2 ) separates b1
from b2 in G, and X is infeasible in Γ.
2.3
Reductions
In some cases, we can show that the desired paths exist in the graph G if and only if certain
disjoint paths exist in some subgraph of G. We define these cases as reductions. Refer to
Figure 10 for the following definition.
Definition 2.10. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph, and let X ⊆ V (G)\{a1 , b1 , a2 , b2 }
with |X| ≥ 2. We say that (Γ, X) satisfies property (R1) if G has some separation (H, K)
of order 4 such that
22
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a2
(a)
(b)
Figure 10: Graphs satisfying property (R1)
(i) {a1 , b1 , a2 , b2 } ⊆ V (H) and X ⊆ V (K),
(ii) the vertices in V (H ∩ K) can be labeled as a01 , b01 , a02 , b02 such that
(a) a0i is cofacial with b0j for each i, j ∈ {1, 2},
(b) any four disjoint paths from {a1 , b1 , a2 , b2 } to {a01 , b01 , a02 , b02 } join ai to a0i and bi
to b0i respectively (i = 1, 2)
Among graphs with 4-separations satisfying condition (i) of property (R1), we can further characterize those graphs which satisfy condition (ii) by the following lemma.
Lemma 2.8. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C. Suppose G has
some separation (H, K) of order 4 such that {a1 , b1 , a2 , b2 } ⊆ V (H), and the vertices in
V (H ∩ K) can be labeled as a01 , b01 , a02 , b02 such that a0i is cofacial with b0j for each i, j ∈ {1, 2},
and any four disjoint paths from {a1 , b1 , a2 , b2 } to {a01 , b01 , a02 , b02 } join ai to a0i , and bi to b0i
(i = 1, 2) respectively. Then one of the following is true:
(i) G has some separation (H 0 , K 0 ) of order at most 4 such that {a1 , b1 , a2 , b2 } ⊆ V (H 0 )
and {a01 , b01 , a02 , b02 } ⊆ V (K 0 ), with {a1 , b1 , a2 , b2 } * V (H 0 ∩ K 0 ) and {a01 , b01 , a02 , b02 } *
V (H 0 ∩ K 0 ); or
(ii) a0i = ai for some i ∈ {1, 2} or b0i = bi for some i ∈ {1, 2}; or
23
(iii) for some i ∈ {1, 2}, either b3−i , b03−i , a0i , ai or ai , a0i , b0i , bi occur in this clockwise order
on C.
This is simply a recasting of Theorem 2.1 in [20].
Notice also that if such a graph G has some separation (H, K) of order 4 such that
{a1 , b1 , a2 , b2 } ⊆ V (H) and one of (ii) or (iii) of Lemma 2.8 is satisfied, then V (H ∩ K)
can in fact be labeled as a01 , b01 , a02 , b02 such that every set of four disjoint paths from A to
V (H ∩ K) consists of paths from ai to a0i and from bi to b0i (i ∈ {1, 2}).
Proposition 2.9. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph, and X ⊆ V (G) \ {a1 , b1 , a2 , b2 }
with |X| ≥ 2. If there is some edge of G incident with ai and bj , i, j ∈ {1, 2}, then (Γ, X)
satisfies property (R1).
Proof. Suppose there is an edge e = ai bj ∈ E(G). Then we have a separation (H, K)
defined by V (H) = {a1 , a2 , b1 , b2 }, E(H) = e, and K = G − e.
We also consider the case where a disk graph has a separation (H, K) similar to that in
property (R1), but the vertex set X need not be contained in V (K). Refer to Figure 11 for
the following definition.
Definition 2.11. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph, let C be the outer cycle of G,
and let X ⊆ V (G) \ {a1 , b1 , a2 , b2 } with |X| ≥ 2. We say (Γ, X) satisfies property (R2) if
there exist some i ∈ {1, 2}, a partition of X into X1 = X ∩ C(b3−i , ai ), X2 = X ∩ C(ai , bi ),
and X3 = X \ (X1 ∪ X2 ), with |X3 | ≥ 2, and a separation (H, K) of G of order at most 4
such that one of the following holds:
(i) X1 , X2 are nonempty, (H, K) is of order at most 3, X1 ∪ X2 ∪ {a3−i , b1 , b2 } ⊆ V (H),
X3 ∪ {ai } ⊆ V (K), and |V (H ∩ K) − V (C[b3−i , bi ])| = 1 (Figure 11(a)); or
(ii) X2 = ∅, X1 ∪{a1 , b1 , a2 , b2 } ⊆ V (H), X3 ⊆ V (K), and |V (H ∩K)−V (C[b3−i , ai ])| = 2
(Figure 11(b)); or
(iii) X1 = ∅, X2 ∪ {a1 , b1 , a2 , b2 } ⊆ V (H), X3 ⊆ V (K), and |V (H ∩ K) − V (C[ai , bi ])| = 2
24
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(a)
(b)
Figure 11: Graphs satisfying property (R2)
We call a disk graph and vertex set (Γ, X) reducible if (Γ, X) satisfies property (R1) or
property (R2).
Proposition 2.10. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph, and let X ⊆ V (G)\{a1 , b1 , a2 , b2 }
such that (Γ, X) is reducible. Assume the separation (H, K) is chosen to maximize K. Then
X is feasible in Γ if and only if there are three disjoint paths in K: from a01 to X, from a02
to X, and from b01 to b02 .
In particular, if Γ0 = (K, a01 , b01 , a02 , b02 ) is a disk graph with X ∩ {a01 , b01 , a02 , b02 } = ∅, then
X is feasible in Γ if and only if X is feasible in Γ0 .
Proof. Suppose there are disjoint paths P1 from a1 to X and P2 from a2 to X. We may
assume there are disjoint paths Q1 from b1 to b01 and Q2 from b2 to b02 , each disjoint from
P1 ∪ P2 . For any such collection of paths, we must have a01 ∈ V (P1 ) and a02 ∈ V (P2 ). So
V (P1 ) ∪ V (P2 ) ∪ {b01 , b02 } separates b1 from b2 in G. Hence X is feasible in Γ if and only if
there is a path from b01 to b02 in K, disjoint from P1 ∪ P2 .
Lemma 2.11. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph, and let X ⊆ V (G) \ {a1 , b1 , a2 , b2 }
with |X| ≥ 2. Suppose G has some separation (H, K) of order 4 such that {a1 , b1 , a2 , b2 } ⊆
V (H) and X ⊆ V (K). Suppose further that V (H ∩ K) can be labeled {k1 , h1 , k2 , h2 } in
clockwise order such that there are (possibly trivial) disjoint paths in K, from k1 to X, from
k2 to X, and from h1 to h2 . Then one of the following holds.
25
(i) X is feasible in Γ, or
(ii) (Γ, X) is reducible, or
(iii) (Γ, X) ∈ O2 .
Proof. If there are not four disjoint paths from {a1 , b1 , a2 , b2 } to V (H ∩ K), then there must
be some separation (G1 , G2 ) of G of order at most three with K ⊆ G1 and {a1 , b1 , a2 , b2 } ⊆
V (G2 ), and hence (Γ, X) ∈ O2 . So we may assume there are four disjoint paths from
{a1 , b1 , a2 , b2 } to V (H ∩ K).
Suppose there is a set of four such paths in which {a1 , a2 } are joined to {k1 , k2 }, so
{b1 , b2 } are joined to {h1 , h2 }. Together with the paths in K, these show that X is feasible
in Γ. We may assume then that the vertices in V (H ∩ K) can be labeled as a01 , b01 , a02 , b02
such that any four disjoint paths from {a1 , b1 , a2 , b2 } to {a01 , b01 , a02 , b02 } join ai to a0i and bi to
b0i respectively (i = 1, 2), where a1 = hj , a2 = h3−j , b1 = kj , b2 = k3−j for some j ∈ {1, 2}.
So (Γ, X) satisfies property (R1).
2.4
Main Theorem
Theorem 1. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph, and let X ⊆ V (G) \ {a1 , b1 , a2 , b2 }
with |X| ≥ 2. Then one of the following holds.
(C1) X is feasible in Γ,
(C2) (Γ, X) is reducible, or
(C3) (Γ, X) ∈
S7
i=1 Oi .
26
CHAPTER III
MAXIMIZING PATHS
3.1
Defining a maximizing path
In this section, we define the choice of a “best” path P from {a1 , a2 } to X in a disk graph.
We then show that this choice greatly restricts the location of the vertices of X within
G − V (P ). Given any path P from {a1 , a2 } to X, we use P to define a separation of G
into “lower” and “upper” subgraphs. We use these subgraphs throughout the proofs which
follow.
Given T ⊆ V (G), a subgraph B of G is a T-bridge of G if either (i) B is induced by
a single edge xy ∈ E(G) \ E(T ) where x, y ∈ V (T ) or (ii) ∃ some component H of G − T
such that B is the subgraph induced by E(H) ∪ {xy ∈ E(G) : x ∈ V (T ), y ∈ V (H)}. The
vertices in V (B) ∩ V (T ) are called the attachments of B in T .
Definition 3.1. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C, and let
X ⊆ V (G) \ {a1 , b1 , a2 , b2 } with |X| ≥ 2. Let P be some path from ai ∈ {a1 , a2 } to X
in G which is disjoint from C[bi , b3−i ]. We denote by LP the minimal union of blocks of
G − V (P ) such that C[bi , b3−i ] ⊆ LP , and note that LP is a chain of blocks (see Figure 12).
By the definition of LP , each (LP ∪ P )-bridge of G has at most one attachment in LP .
Since G is 2-connected, each (LP ∪ P )-bridge has at least one attachment in P . Hence
G − LP is connected, and so there is a unique LP -bridge of G. Let UP be this LP -bridge.
Let MP denote the path induced by the edges in E(∂LP ) − E(C[bi , b3−i ]).
Definition 3.2. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph, let X ⊆ V (G) \ {a1 , b1 , a2 , b2 }
with |X| ≥ 2, and let P be some path from ai ∈ {a1 , a2 } to x0 ∈ X in G which is disjoint
from C[bi , b3−i ].
For vertices u, v ∈ MP we say u lies to the left of v if v ∈ MP (u, b1 ], in which case we
also say v lies to the right of u.
27
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1
a2
Figure 12: The subgraph LP (shaded), where P is from a1 to x
Given a path Q from q 0 ∈ V (P ) to q ∈ V (MP ), internally disjoint from P ∪ LP , we
say Q falls to the left of P if all vertices of MP − q which are cofacial with any vertex of
P (q 0 , x0 ] lie to the right of q. Symmetrically, we say Q falls to the right of P if all vertices
of MP − q which are cofacial with any vertex of P (q 0 , x0 ] lie to the left of q.
If D is a facial cycle of G with D ∩ P 6= ∅ and D ∩ LP 6= ∅, then there are two paths
from P to MP which are subgraphs of D. We say the face bounded by D is left of P if both
of these paths fall to the left of P , and we say it is right of P if both of these paths fall to
the right of P .
We will use paths from P to LP extensively, so let us take a moment to observe their
character. Suppose we have such a path S from s0 ∈ V (P ) to s ∈ V (MP ). Then S is
certainly contained in some (LP ∪ P )-bridge of G. G is (3, {a1 , b1 , a2 , b2 })-connected, so if
S has any internal vertices, the bridge must have at least two attachments on P . So a path
from P to LP is either a single edge, or it is contained in some bridge with at least two
attachments in P . We sometimes refer to a bridge which is a single edge from P to LP as
a trivial bridge, and a bridge with at least one internal vertex as non-trivial.
Suppose (Γ, X) satisfies the definition of O3 for fixed i ∈ {1, 2}. That is, for each x ∈ X,
one of the following holds:
(i) x ∈ C[b3−i , bi ], or
(ii) there is a finite face of G incident with both ai and x, or
28
(iii) G has a separation (H, K) of order 3 such that ai ∈ V (H∩K), x ∈ V (K), {a1 , a2 , b1 , b2 } ⊆
V (H)
We say (Γ, X) ∈ O3 with respect to ai . This distinction allows us to state the following
lemma more exactly, and will thus simplify several later proofs.
Lemma 3.1. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph, and let X ⊆ V (G) \ {a1 , b1 , a2 , b2 }
with |X| ≥ 2. Then for each i ∈ {1, 2}, either (Γ, X) ∈ O3 with respect to a3−i , or
G − (C[bi , b3−i ]) contains a path P from ai to X for which a3−i is not a cutvertex of LP .
Proof. Given i ∈ {1, 2}, let H = G − a3−i . Observe that since G is 2-connected, H is
connected. Since G is (3, {a1 , b1 , a2 , b2 })-connected, H is a chain of blocks, with bi and
b3−i lying in the endblocks of H. Define D to be the path in H induced by the edges in
E(∂H) − E(C[b3−i , bi ]).
Suppose no desired path P from ai to X exists. Then for each xj ∈ X, there is no
path from ai to xj in G − D. Hence by planarity either xj ∈ V (D) or there is some
separation (Hj , Mj ) of H of order 2 such that V (Hj ∩ Mj ) ⊆ V (D), xj ∈ V (Mj ), and
{ai , b1 , b2 } ⊆ V (Hj ).
If xj ∈ V (D), then xj is cofacial with a3−i . Where the separation (Hj , Mj ) exists, let Hj0
be the subgraph of G induced by V (Hj ) ∪ {a3−i }, and let Mj0 be the union of all Hj0 -bridges
of G. We then have a3−i ∈ V (Hj0 ∩ Mj0 ), xj ∈ V (Mj0 ), {a1 , a2 , b1 , b2 } ⊆ V (Hj0 ), and we see
that (Γ, X) ∈ O3 with respect to a3−i .
Definition 3.3. Given Γ = (G, a1 , b1 , a2 , b2 ) a disk graph, and given X ⊆ V (G)\{a1 , b1 , a2 , b2 }
with |X| ≥ 2, we let P(Γ,X) be the collection of all paths P from ai ∈ {a1 , a2 } to X in
G − C[bi , b3−i ] such that a3−i is not a cutvertex of LP .
Note from Lemma 3.1 that if (Γ, X) ∈
/ O3 , then P(Γ,X) 6= ∅.
Definition 3.4. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph, and let X ⊆ V (G)\{a1 , b1 , a2 , b2 }
with |X| ≥ 2. For a path P ∈ P(Γ,X) from ai to X in G, let
R(P ) = {v ∈ V (LP ) \ V (MP ) : ∃ a path from a3−i to v in LP − V (MP )},
A(P ) = {v ∈ V (LP ) : v lies in the block of LP containing a3−i }, and
L(P ) = {v ∈ V (G) : v lies in the same component of G − V (P ) as LP }.
29
a1
1
0
1
0
ua 1010
B
0
1
1
0
0
1
S
00
11
11
00
00
11
ux
1
0
0
1
0
1
x1
00
11
11
00
00
11
00
11
11
00
00
11
0
1
0
1
b210
b1
1
0
0
1
0
1
a2
Figure 13: Bridges without LP attachment
Suppose we have a path P ∈ P(Γ,X) from {a1 , a2 } to X, and there is some x2 ∈ X ∩R(P ).
Let Q be some path from a2 to x2 in LP disjoint from MP . Note that MP provides a path
from b1 to b2 which is disjoint from P ∪ Q, so we see that X is feasible in Γ. We can think of
R(P ) as the set of “reachable” vertices associated with the path. If there is some reachable
vertex in X, we will have the desired disjoint paths. Our approach, then, is to choose a path
which maximizes the set of reachable vertices, and then examine the potential locations of
vertices in X within the subgraph LP .
Definition 3.5. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph, and let X ⊆ V (G)\{a1 , b1 , a2 , b2 }
with |X| ≥ 2. We say that a path P ∈ P(Γ,X) is a maximizing path if, among all paths in
P(Γ,X) , considered under set inclusion,
(i) R(P ) is maximal,
(ii) subject to (i), A(P ) is maximal,
(iii) subject to (ii), V (LP ) is maximal, and
(iv) subject to (iii), L(P ) is maximal.
Lemma 3.2. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph, and let X ⊆ V (G) \ {a1 , b1 , a2 , b2 }
with |X| ≥ 2. Let P ∈ P(Γ,X) be a maximizing path. Then every (P ∪ LP )-bridge of G has
exactly one attachment in LP .
30
Proof. Without loss of generality, assume P is from a1 to x1 ∈ X. (See Figure 13.) By the
definition of LP , each (LP ∪ P )-bridge of G has at most one attachment in LP .
Let B1 , . . . , Bn be the (P ∪ LP )-bridges of G which have no attachments in LP . For
each Bk , let Pk be the shortest subpath of P including all attachments of Bk in P . Since
G is 2-connected, Bk ∪ Pk must be 2-connected. For any K ⊆ {1, 2, . . . , n}, define BK =
S
k∈K (Bk ∪ Pk ). Choose a maximal J ⊆ {1, 2, . . . , n} such that BJ is 2-connected. Let
B := BJ , and let ua , ux be the vertices of B ∩ P which lie closest along P to a1 and x1 ,
respectively.
Since G is (3, {a1 , b1 , a2 , b2 })-connected, we see that {ua , ux } is not a 2-cut in G, so there
must be some path S from s0 ∈ P (ua , ux ) to s ∈ P [a1 , ua )∪P (ux , x1 ]∪LP , internally disjoint
from P ∪ LP . We claim that we may choose such a path with s ∈ V (LP ). Otherwise, each
such path S is contained in some (P ∩ LP )-bridge Bj of G which has no attachment in LP .
Then B ∪ Bj is 2-connected, and J ∪ {j} contradicts the maximality of J.
Note that either s0 ∈ ∂B(ux , ua ) or s0 ∈ ∂B(ua , ux ). Suppose without loss of generality
that s0 ∈ ∂B(ux , ua ) (see Figure 13). We form the new path Q from P by replacing
P [ua , ux ] by ∂B[ua , ux ]. Some vertices of B − V (Q) now lie in a (Q ∪ LQ )-bridge of G with
attachment s in LQ . Note that R(P ) ⊆ R(Q), A(P ) ⊆ A(Q), LP ⊆ LQ , and L(P ) ⊆ L(Q),
but L(P ) 6= L(Q). This contradicts the choice of P as a maximizing path, and hence cannot
occur.
The following lemma applies a similar argument to limit the type of paths which may
exist within (P ∪ LP )-bridges. Lemmas 3.3 and 3.4 are illustrated in Figure 14.
Lemma 3.3. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph, and let X ⊆ V (G) \ {a1 , b1 , a2 , b2 }
with |X| ≥ 2. Let P ∈ P(Γ,X) be a maximizing path, and assume P is from a1 to x1 ∈ X.
Suppose there is a path W from w0 ∈ V (P ) \ {x1 } to w ∈ P (w0 , x1 ] which is internally
disjoint from P ∪ LP . Then all paths from P (w0 , w) to MP which are internally disjoint
from P ∪ LP ∪ W have a common endpoint in MP .
Proof. Let S be a path from s0 ∈ P (w0 , w) to s ∈ V (MP ) and let R be a path from
r 0 ∈ P (w0 , w) to r ∈ V (MP ), such that S and R are each internally disjoint from P ∪LP ∪W .
31
a1
11
00
00
11
W
00
11
11
00
00
11
R
S
b2
11
00
00
11
11
00
00
11
b1
1
0
0
1
11
00
00
11
00
11
a2
Figure 14: Where b1 , b2 ⊆ V (UP )
Suppose that s 6= r.
Let Q := P [a1 , w0 ] ∪ W ∪ P [w, x1 ]. Now (S ∪ R) ⊆ LQ . Note that R(P ) ⊆ R(Q),
A(P ) ⊆ A(Q), and V (LP ) ⊆ V (LQ ), but V (LP ) 6= V (LQ ). This contradicts the choice of
P as a maximizing path.
Lemma 3.4. Let Γ = (G, a1 , b1 , a2 , b2 ), and let X ⊆ V (G) \ {a1 , b1 , a2 , b2 } with |X| ≥ 2.
Let P ∈ P(Γ,X) be a maximizing path, and assume P is from a1 to x1 . If X \ {x1 } ⊆ V (LP )
and {b1 , b2 } ⊆ V (UP ), then (Γ, X) is reducible.
Proof. Suppose b1 , b2 ∈ V (UP ). If any vertex v ∈ P (a1 , x1 ] is cofacial with each of b1 , b2 , we
have a 4-separation (H, K) of G with V (H ∩ K) = {v, b1 , a2 , b2 }, {a1 , b1 , a2 , b2 } ⊆ V (H),
and X ⊆ V (K). Hence (Γ, X) satisfies property (R1). We may assume, then, that no such
vertex exists. So there is some path from P [a1 , x1 ) to MP (b2 , b1 ). Let S be such a path,
from s0 ∈ P [a1 , x1 ) to s ∈ MP (b2 , b1 ), chosen to minimize P [a1 , s0 ]. Assume by symmetry
that S falls to the right of P , so s0 is cofacial with b1 .
Now s0 is not cofacial with b2 . From our choice of S, there is no path from P [a1 , s0 ) to
MP (b2 , b1 ), so there is some path W from w0 ∈ P [a1 , s0 ) to w ∈ P (s0 , x1 ], internally disjoint
from P . Choose such a path to minimize P [a1 , w0 ], so w0 is cofacial with b2 .
Now w0 is not cofacial with b1 . Again considering our choice of S, we must have some
path U from u0 ∈ P [a1 , w0 ) to u ∈ P (w0 , x1 ], internally disjoint from P . By Lemma 3.2, this
must be contained in a (P ∪ LP )-bridge with some attachment in MP . Label this bridge
32
Y , and let y ∈ MP [s, b1 ] be its attachment in LP . There is some path from u0 to y in Y ,
internally disjoint from P . By our choice of S, we must have y = b1 . There is also a path
from u to y in Y , internally disjoint from P , but this contradicts Lemma 3.3.
3.2
Examining path bridges
Here we begin with a maximizing path P , and consider whether X ⊆ LP . We find that all
vertices of X \ V (LP ) must lie in a specific obstructive relationship.
Lemma 3.5. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph, and let X ⊆ V (G) \ {a1 , b1 , a2 , b2 }
with |X| ≥ 2. Let P ∈ P(Γ,X) be a maximizing path, and assume P is from a1 to x1 ∈ X.
Let Y = X \ V (LP ). Then either
1. Y = {x1 }, or
2. G has a facial cycle D and a 3-separation (H, K) with V (H ∩ K) = {u, r, w} such
that
(a) u ∈ V (P ), {r, w} ⊆ V (D ∩ MP ), and D[w, r] = MP [r, w],
(b) {a1 , b1 , a2 , b2 } ⊆ V (H), Y ⊆ V (K), and
(c) Y ⊆ D[r, w].
In case (2), (Γ, Y ) ∈ O1 ∩ O2 .
Proof. Suppose Y 6= {x1 }. Label Y = {x1 , y1 , . . . , ym }. For each yi ∈ Y , let Bi denote the
(P ∪ LP )-bridge of G containing yi . Let ui , vi be the attachments of Bi in P chosen to
minimize each of P [a1 , ui ] and P [vi , x1 ]. Now choose j ∈ {1, . . . , m} to minimize P [a1 , uj ].
(See Figure 15.)
By Lemma 3.2, Bj must have a unique attachment in LP . Let w be this attachment.
Since G is (3, {a1 , b1 , a2 , b2 })-connected, {uj , w} is not a 2-cut in G. This shows uj 6= vj ,
and that there is some path from P (uj , x1 ] to LP ∪ P [a1 , uj ) in G − {uj , w}. From Lemma
3.2 we see that G in fact has some path R from r 0 ∈ P (uj , x1 ] to r ∈ V (LP ) which is
internally disjoint from P ∪ LP ∪ Bj .
33
a1
11
00
00
11
11
00
00
11
R
r
1
0
1
0
b2
Bj
11
00
11
00
1
0
0
1
x1 y j
D
11
00
00
11
00
11
11
00
11
00
00
11
11
00
00
11
w
11
00
00
11
11
00
00
11
b1
11
00
00
11
00
11
a2
Figure 15: Where X \ {x1 } is not contained in V (LP )
Suppose {uj , w, r} is not a 3-cut in G. Then there is some path S from P (uj , x1 ] to LP
which is disjoint from r and is internally disjoint from P ∪ LP ∪ Bj . Now Bj contains a path
R0 from uj to yj which is internally disjoint from P ∪ LP . Let Q = P [a1 , uj ] ∪ R0 . Then
R(P ) ⊆ R(Q), A(P ) ⊆ A(Q), and (R ∪ S ∪ LP ) ⊆ LQ , contradicting the choice of P as a
maximizing path.
Hence we may assume T := {uj , w, r} is a 3-cut in G. Let K be the T -bridge of
G containing yj . Then we have a separation (H, K) of G with V (H ∩ K) = T and
{a1 , b1 , a2 , b2 } ⊆ V (H). Since P (uj , x1 ] ⊆ K, we have x1 ∈ V (K). From our choice of
j, and from planarity, we see that Y ⊆ V (K). So condition (b) is satisfied by this separation.
Now there is some finite face of G bounded by a cycle D such that D ∩ LP = D[w, r].
Suppose there is some path P1 from uj to Y in K which is disjoint from D[r, w]. Letting
Q := P [a1 , uj ]∪P1 , we see that D[r, w] ⊂ LQ , so we have R(P ) ⊆ R(Q), A(P ) ⊆ A(Q), and
V (LP ) ⊆ V (LQ ), but V (LP ) 6= V (LQ ). This contradicts the choice of P as a maximizing
path, and hence cannot occur, so there can be no such path P1 .
From planarity, we see that for each y ∈ Y , either y ∈ D[r, w] or there is some 2-cut in
D[r, w] separating y from {uj , r, w}. Since G is (3, {a1 , b1 , a2 , b2 })-connected, no such 2-cut
exists. Hence Y ⊆ D[r, w], and conditions (a) and (c) are satisfied.
Let us state as a corollary the special case of this lemma where |X| = 2.
34
Corollary 3.6. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph, and let X = {x1 , x2 } ⊆ V (G) \
{a1 , b1 , a2 , b2 } with |X| ≥ 2. Let P ∈ P(Γ,X) be a maximizing path, and assume P is from
a1 to x1 . Then either x2 ∈ V (LP ) or (Γ, X) ∈ O1 ∩ O2 .
3.3
Location of X relative to a
Now let us suppose we have a maximizing path from a1 to x1 , and X 0 := X \{x1 } ⊆ V (LP ).
We show that X 0 cannot be separated from a2 by a cutvertex in LP . That is, the minimal
chain of blocks in LP containing X 0 must include the block containing a2 . It can actually be
shown that under the conditions of Lemma 3.7, there is some x2 ∈ X ∩ A(P ). This weaker
result will be enough for our purposes, however, and the proof is more straightforward.
Lemma 3.7. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C. Let X ⊆
V (G) \ {a1 , b1 , a2 , b2 }, with |X| ≥ 2, X ∩ V (C) = ∅, and X infeasible in Γ. Let P ∈ P(Γ,X)
be a maximizing path, and assume P is from a1 to x1 . Let X 0 = X \ {x1 }. Suppose
X 0 ⊆ V (LP ) and there is some cutvertex of LP separating X 0 from a2 . Then either (Γ, X)
satisfies property (R1) or (Γ, X) ∈ O1 ∪ O2 ∪ O3 ∪ O5 .
Proof. Label the unique block of LP containing a2 as Ba . Label the minimal chain of blocks
containing X 0 as Bx . Since there is a cutvertex of LP separating X 0 from a2 , |Ba ∩ Bx | ≤ 1.
We may assume without loss of generality that Bx and Ba appear in clockwise order along
C[b1 , b2 ]. See Figures 16 and 17 for illustrations.
Let {y1 , y2 } = V (LP ∩ UP ∩ C), such that C[b1 , b2 ] ⊆ C[y1 , y2 ]. If b1 ∈
/ Bx , let z0 be the
cutvertex of LP separating X 0 from b1 which appears closest to y1 along MP , and let z1 be
the cutvertex of LP which separates X 0 from b1 with z1 ∈ V (Bx ) (z0 = z1 is possible). If
b1 ∈ Bx , let z0 = z1 = b1 . In either case, let z2 be the cutvertex of LP separating a2 from X 0
with z2 ∈ V (Bx ), and let z3 be the cutvertex of LP separating a2 from X 0 with z3 ∈ V (Ba )
(z2 = z3 is possible). If y2 ∈ V (Ba ), let z4 = y2 , otherwise let z4 be the cutvertex of LP
separating a2 from b2 with z4 ∈ V (Ba ). Finally, if no block of LP includes both b2 and z4 ,
let z5 be the cutvertex of LP separating them which appears closest to z4 along MP . If
some block contains both b2 and z4 , let z5 = b2 .
35
a1
1
0
0
1
0
1
0
1
0
1
0
w001
W0
W
1
0
0
1
y2
S0
S
T
1
0
1
0
1
0
0
1
0
1
z5 = b2
y1
b1
z0
11
00
00
11
00
11
z4
00
11
00
11
00
11
11
00
00
11
1
0
0
1
0
1
00
11
11
00
00
11
0
1
0
1
0
1
B
0a
1
0
1
0
1
a2
1
0
0
1
1
0
0
1
z2
z3
Bx
z1
Figure 16: Lemma 3.7, Claim 1
Since G is (3, {a1 , b1 , a2 , b2 })-connected, the set {z1 , z2 } cannot be a 2-cut in G. Hence
there is some path S from s ∈ MP (z2 , z1 ) to s0 ∈ V (P ) in G. We choose such a path S to
minimize P [a1 , s0 ].
Note that if there is a separation (M, K) of G with V (M ∩K) = {s0 , z1 , z2 }, {a1 , b1 , a2 , b2 } ⊆
V (M ) and X ⊆ V (K), we have (Γ, X) ∈ O2 . Hence we may assume there is no such separation, and by our choice of S we have s0 6= x1 .
Claim 1. We may assume that there exist two disjoint paths from P (s0 , x1 ] to MP [b2 , z2 ) ∪
MP (z1 , b1 ].
Proof of Claim 1 If there is a 4-separation (H, K) of G such that {a1 , b1 , a2 , b2 } ⊆ V (H),
X ⊆ V (K), and {z1 , z2 , s0 } ⊆ V (H ∩ K), then (Γ, X) satisfies property (R1), since z1 , z2 ∈
C[b1 , a2 ]. We may therefore assume no such separation exists, and hence by the choice of
S, there must be two disjoint paths from P (s0 , x1 ] to MP [b2 , z2 ) ∪ MP (z1 , b1 ] ∪ P [a1 , s0 ).
Applying Lemma 3.2, we may assume that at least one of these paths ends in MP .
Suppose Claim 1 fails. Let T be a path from t0 ∈ P (s0 , x1 ] to t ∈ MP [b2 , z2 ) ∪ MP (z1 , b1 ]
and let W 0 be a path from w0 ∈ P (s0 , x1 ] to w00 ∈ P [a1 , s0 ) with T and W 0 disjoint (Figure
16). W 0 must lie in some (P ∪ LP )-bridge of G. From Lemma 3.2, this bridge has some
attachment w in LP . From the planarity of G and the choice of T , we see that w ∈
MP [b2 , z2 ) ∪ MP (z1 , b1 ], and hence there is some path W from w0 to w internally disjoint
36
a1
1
0
0
1
0
1
T
y2
W
1
0
0
1
0
1
S0
x1
S
1
0
1
0
00
11
00
00
11
z5 = b211
z
4
11
00
00
11
00
11
B
0a
1
T0
z3
z
z2
1
0
0
1
1
0
0
1
0
1
0
1
z11
1
00
0
11
00
00
11
00
11
00
11
00
11
00
11
11
00
00
11
00
11
y1
b1
11
00
00
11
Bx
a2
Figure 17: Lemma 3.7, Claim 2
from P ∪ LP .
Now for any path R from r 0 ∈ P (s0 , x1 ] to r ∈ MP [b2 , z2 ) ∪ MP (z1 , b1 ], R is not disjoint
from W . Since we have t0 6= w0 , it must be that for all such paths R, r = w.
If w00 is cofacial with one of z1 or z2 , then we have a separation (H, K) of G with
V (H ∩ K) = {w00 , z1 , z2 , w}, {a1 , b1 , a2 , b2 } ⊆ V (H), and X ⊆ V (K), and we find (Γ, X)
satisfies property (R1). So we may assume w00 is not cofacial with z1 or z2 . By our choice
of S, we find that there must be some path S0 from P (w00 , s0 ] to MP [b2 , z2 ) ∪ MP (z1 , b1 ].
Now let Q := P [a1 , w00 ] ∪ W 0 ∪ P [w0 , x1 ]. Note that (S ∪ S0 ) ⊆ LQ , so we have R(P ) ⊆
R(Q), A(P ) ⊆ A(Q), V (LP ) ⊆ V (LQ ), but V (LP ) 6= V (LQ ), contradicting the assumption
that P is a maximizing path. This proves Claim 1.
By Claim 1, let T and W be disjoint paths from t0 ∈ P (s0 , x1 ] to t ∈ MP [b2 , z2 )∪MP (z1 , b1 ]
and w0 ∈ P (s0 , x1 ] to w ∈ MP [b2 , z2 ) ∪ MP (z1 , b1 ] respectively, with s0 , t0 , w0 , x1 appearing
in order on P (w0 = x1 is possible). Assume that T is chosen to minimize P [s0 , t0 ], and
subject to this to maximize MP [t, s]. From planarity, we have either {t, w} ⊆ MP [y2 , z2 ) or
{t, w} ⊆ MP (z1 , y1 ].
Claim 2. We may assume that t, w ∈ MP [y2 , z2 ).
Proof of Claim 2
Suppose on the contrary that t, w ∈ MP (z1 , y1 ], as illustrated in Figure
17. Note that this can occur only if b1 ∈
/ V (Bx ). By planarity, S is to the left of P , and
37
a1
1
0
0
1
S
T
x1
y11
2
00
11
00
00
11
11
00
11
00
b2
z
4
1
0
0
1
0
1
z0
W0
W
z5
1
0
0
1
0
1
1
0
0
1
0
1
z3
z2
1
0
0
1
1
0
0
1
11
00
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00
z1
0
1
0
1
1
0
0
1
0 1
1
0
0
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1
0
00
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00
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00
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1
0
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1
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1
0
0
1
y1
b1
1
0
0
1
0
1
Bx
a2
Figure 18: Lemma 3.7, choice of path W
hence no path from P (s0 , x1 ] to MP [y2 , z2 ] exists. We may further assume S is chosen with
MP [z2 , s] minimal. Therefore z2 and s0 are cofacial.
We may assume (Γ, X) does not satisfy property (R1), so {z2 , s0 , z0 } cannot be three
vertices in a 4-cut in G. Hence T can be chosen such that t ∈ MP (z0 , y1 ]. Also since
(Γ, X) does not satisfy property (R1), {z2 , s0 , y1 , b1 } cannot be a 4-cut in G, so there must
be some path T0 from t00 ∈ P [a1 , s0 ) to t0 ∈ MP [t, y1 ). Choose such a path to minimize
P [a1 , t00 ] and MP [t0 , y1 ]. Then t0 , t00 , and y1 are cofacial. Note from our choice of z0 that
MP [t, t0 ] ∩ C[b1 , b2 ] = ∅. Now {z3 , t00 , y1 , b1 } cannot be a 4-cut in G, since (Γ, X) does not
satisfy property (R1), so there must be some path S0 from s00 ∈ P (t00 , s0 ] to s0 ∈ MP [y2 , z3 ).
Let Q := P [a1 , t00 ] ∪ T0 ∪ MP [t, t0 ] ∪ T ∪ P [t0 , x1 ]. Then paths S0 and S demonstrate
R(P ) ⊆ R(Q), A(P ) ⊆ A(Q), and A(P ) 6= A(Q). This contradicts the assumption that P
is a maximizing path, and hence proves the claim.
By Claim 2 and planarity, no path from P (s0 , x1 ] to MP [z1 , y1 ] exists, and S lies to the
right of P (as in Figure 18). We may assume S is chosen with MP [s, z1 ] minimal, so z1 and
s0 are cofacial. Since G is (3, {a1 , b1 , a2 , b2 })-connected, s must belong to the endblock Z of
Bx containing z1 , and s is not a cutvertex of Bx . Since X ∩ V (C) = ∅, Z contains a vertex
of X which is not a cutvertex of Bx . So there is a path S 0 from s to X 0 in Bx which is
disjoint from C[b1 , b2 ].
If {z1 , z3 , s0 } is a 3-cut in G separating X from {a1 , b1 , a2 , b2 }, we see that (Γ, X) ∈ O2 ,
38
a1
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0
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T
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y11
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1
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a2
Figure 19: Lemma 3.7, Claim 4
so we may assume that it is not. Therefore, t ∈ MP [y2 , z3 ). Assume further that W is
chosen to minimize MP [w, z2 ].
Claim 3. Suppose W0 is a path from w00 ∈ P (s0 , x1 ] to w0 ∈ MP (z4 , s). Then w0 ∈ MP [z2 , s]
and for any choice of the path S 0 from s to X 0 in Bx , disjoint from C[b1 , b2 ], we have
V (S 0 ) ∩ V (MP [z2 , w0 ]) 6= ∅.
Proof of Claim 3 Let W0 be a path from w00 ∈ P (s0 , x1 ] to w0 ∈ MP (z4 , s). If w0 is
left of z2 , let MW = MP [w0 , z2 ]. If w0 is right of z2 , let MW := MP [z2 , w0 ]. Suppose,
contrary to our claim, that there is some choice of path S 0 with V (S 0 ) ∩ V (MW ) = ∅ (as
illustrated in Figure 18). Then there is a path Q := P [a1 , s0 ] ∪ S ∪ S 0 which is disjoint from
C[b1 , b2 ] ∪ T ∪ P [t0 , w0 ] ∪ W0 ∪ MW . The paths T and W0 demonstrate that R(P ) ⊆ R(Q)
and A(P ) ⊆ A(Q), but A(P ) 6= A(Q) contradicting the choice of P as a maximizing path.
This proves Claim 3.
By Claim 3 and our choice of W , there is no path from P (s0 , x1 ] to MP (z4 , s), and so
w ∈ MP (t, z4 ]. From planarity and t 6= w, we find t ∈ MP [y2 , z4 ).
Claim 4. We may assume there is a facial cycle D of G with MP [z3 , z4 ] ∪ W ⊆ D, and
hence {z3 , z4 } is a 2-cut in G.
39
Proof of Claim 4
Suppose there is no such facial cycle. By Claim 3, we see that there
must be some path S0 from s0 to s0 ∈ MP [z4 , z3 ]. (See Figure 19.)
If s0 = a1 , we have 3-cuts {z1 , z2 , a1 } and {z4 , b2 , a1 } demonstrating that Γ ∈ O3 . Hence
we may assume s0 6= a1 . Recall that T is chosen to minimize MP [b2 , t]. If b2 lies in the same
block of LP as t, let z = b2 . Otherwise, let z be the cutvertex of LP separating t from b2
which minimizes MP [z, t]. If {z, s0 , z1 } is a 3-cut in G, we have a 4-separation (M, K) of
G with V (M ∩ K) = {z, s0 , z1 , a2 }, {a1 , b1 , a2 , b2 } ⊆ V (M ), and X ⊆ V (K), showing that
(Γ, X) satisfies property (R1). Hence we may assume this is not a 3-cut, and there is some
path T0 from t00 ∈ P [a1 , s0 ) to t0 ∈ MP (z, t].
Let Q = P [a1 , t00 ] ∪ T0 ∪ MP [t0 , t] ∪ T ∪ P [t, x1 ]. Then the paths S0 and S demonstrate
that R(P ) ⊆ R(Q) and A(P ) ⊆ A(Q), but A(P ) 6= A(Q), contradicting the choice of P as
a maximizing path.
So the path S0 must not exist, and we see that all vertices of MP [z3 , z4 ] ∪ W are indeed
incident with some common finite face of G. This proves Claim 4.
We also note by our choices of T and W that z4 6= y2 , and hence b2 ∈
/ Ba .
Claim 5. Let D be the facial cycle of G such that z3 , z4 ∈ V (D) and W ⊆ D. Then either
(i) X 0 ⊆ V (D), or
(ii) G has a separation (H, K) such that (X 0 \ V (D)) ⊆ V (K), {a1 , b1 , a2 , b2 , x1 } ⊆ V (H),
V (H ∩ K) = {z3 , c, v}, c ∈ V (C), and v ∈ V (D).
If s ∈ V (D), then {z1 , z3 , s0 } is a 3-cut in G, and we have the desired separation with
v = s0 and c = z1 . So we assume s0 ∈
/ V (D), so there must be some path R from r 0 ∈ P (s0 , x1 ]
to r ∈ MP (z3 , z1 ). (See Figure 20.)
From our choice of W (minimizing MP [w, z2 ]) and our earlier finding that w ∈ MP (t, z4 ],
we see that r ∈ MP [z2 , z1 ). Together with the (3, {a1 , b1 , a2 , b2 })-connectivity of G, this
shows that either z2 = z3 , or the block of LP containing z2 and z3 is induced by a single
edge z2 z3 ∈ E(G). Choose the path R to minimize MP [z2 , r]. Note that MP [z2 , r] ⊆ D.
By Claim 3, there must not be a path from s to X 0 in Bx − V (MP [z2 , r] ∪ C[b1 , b2 ]).
40
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c
Bx
Figure 20: Lemma 3.7, Claim 5
Since X 0 ∩ V (C) = ∅, for each x ∈ X 0 we have either x ∈ MP [z2 , r] or there is some
separation (H 0 , K 0 ) of Bx with V (H 0 ∩ K 0 ) ⊆ V (MP [z2 , r] ∪ C[b1 , b2 ]), x ∈ V (K 0 ), and
{a1 , b1 , a2 , b2 , x1 } ⊆ V (H 0 ).
If X 0 ⊆ V (MP [z2 , r]) then condition (i) of the claim is satisfied. Hence we may assume
some separation (H 0 , K 0 ) as above exists. From the (3, {a1 , b1 , a2 , b2 })-connectivity of G and
from our choice of R we see V (H 0 ∩ K 0 ) * MP [z2 , r], so we may assume V (H 0 ∩ K 0 ) = {v, c}
with v ∈ V (MP [z2 , r]) and c ∈ V (C[z1 , z2 ]). We further assume that the separation is
chosen to maximize K 0 , giving X 0 \ V (MP [z2 , r]) ⊆ V (K 0 ). Again from the choice of R, we
find that {v, c, z3 } is a 3-cut in G, associated with a separation which satisfies condition (ii)
of the claim. This proves Claim 4.
Let S0 ⊆ D be the path from s00 ∈ P [s0 , w0 ] to s0 ∈ MP [z2 , s] maximizing P [s00 , x1 ] and
MP [z3 , s0 ]. (See Figure 21. S0 = S is possible.) So D = W ∪ P [w0 , s00 ] ∪ S0 ∪ MP [w, s0 ].
Suppose x1 ∈ V (D). If X 0 ⊆ V (D), then we have (Γ, X) ∈ O1 . Otherwise, from Claim
4 we have a separation (H, K) of G such that V (H ∩ K) = {z3 , c, v}, c ∈ C[b1 , b2 ] − a2 ,
z3 , v ∈ V (D), z3 ∈ C(c, a2 ], {a1 , a2 , b1 , b2 } ⊆ V (H), and X ⊆ V (K ∪ D). This gives
(Γ, X) ∈ O7 . Hence we may assume that x1 ∈
/ V (D), so w0 6= x1 .
From (3, {a1 , b1 , a2 , b2 })-connectivity of G, neither the set {s00 , s0 } nor the set {w0 , w} is
a 2-cut in G, so there is some path U from u0 ∈ P (s00 , x1 ] ∩ P (w0 , x1 ] to u ∈ MP [b2 , w) ∪
MP (s0 , s], as shown in Figure 21.
41
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(a) W right of P
a1
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y211
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(b) W left of P
Figure 21: Lemma 3.7 final cases
42
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0
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1
y1
b1
Suppose W lies to the right of P . Here u ∈ MP [b2 , w). If {z4 , z5 , s00 } is a 3-cut in G, we
find Γ ∈ O5 . We assume, then, that this is not a 3-cut, and hence there is some path T0
from t00 ∈ P [a1 , s00 ) to t0 ∈ MP (z5 , u]. Letting Q := P [a1 , t00 ] ∪ T0 ∪ MP [t0 , u] ∪ U ∪ P [u0 , x1 ],
we see that V (D[w, s0 ]) ⊆ A(Q). Hence A(P ) 6= A(Q) although A(P ) ⊆ A(Q), and
R(P ) ⊆ R(Q), contradicting the choice of P as a maximizing path.
Now suppose W lies to the left of P . Here u ∈ MP (s0 , s]. From Claim 4, we see that
X 0 ∩ MP [u, s] = ∅. Let Q := P [a1 , s0 ] ∪ S ∪ MP [u, s] ∪ U ∪ P [u0 , x1 ]. Then V (T ∪ S0 ) ⊆ A(Q).
Hence A(P ) 6= A(Q) although A(P ) ⊆ A(Q), and R(P ) ⊆ R(Q), again contradicting the
choice of P as a maximizing path.
Corollary 3.8. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C. Let X =
{x1 , x2 } ⊆ V (G) \ {a1 , b1 , a2 , b2 }, with x1 , x2 ∈
/ V (C) and X infeasible in Γ. Let P ∈ P(Γ,X)
be a maximizing path, and assume P is from a1 to x1 . Then either x2 ∈ A(P ), or (Γ, X)
satisfies property (R1), or (Γ, X) ∈ O1 ∪ O2 ∪ O3 ∪ O5 .
Proof. From Corollary 3.6, we see that either (Γ, X) ∈ O1 ∩ O2 or x2 ∈ V (LP ), so we
may assume x2 ∈ V (LP ). Now if x2 ∈
/ A(P ), there is some cutvertex of LP separating x2
from a2 . By application of Lemma 3.7, either (Γ, X) satisfies property (R1) or (Γ, X) ∈
O1 ∪ O2 ∪ O3 ∪ O5 .
3.4
Location of X relative to path
Assuming we have a maximizing path P , we now consider the location of X within LP . We
begin with two propositions which illustrate potential path routings. Propositions 3.9 and
3.10 are illustrated in Figure 22.
Proposition 3.9. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph, and let x1 , x2 ∈ V (G) \
{a1 , b1 , a2 , b2 } such that {x1 , x2 } is infeasible in Γ. Let P ∈ P(Γ,{x1 ,x2 }) be a path from
a1 to x1 , with x2 ∈ V (MP ). Suppose we have four paths Ps , Pu , Pw , and Pr from
p0s , p0u , p0w , p0r ∈ V (P ) to ps , pu , pw , pr ∈ V (MP ) respectively, such that
43
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1
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0
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1
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Qt
Pw x 1010
Ps
1
Pu Pr
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Qw Qr
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01
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1
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a2
a2
(a) Proposition 3.9
(b) Proposition 3.10
Figure 22: Establishing separations when special paths exist
1. a1 , p0s , p0u , x1 appear in order along P ,
2. p0w , p0r ∈ P (p0s , p0u ),
3. pw , pu , pr , and x2 are distinct, and ps 6= b1 ,
4. b2 , pw , pu , pr , ps , b1 appear in order along MP , and
5. x2 ∈ MP (pr , b1 ).
Then there is a separation (H, K) of LP of order at most three such that {b2 , a2 , b1 , ps } ⊆
V (H) and {pw , pu , pr , x2 } ⊆ V (K).
Proof. Given such a paths (as in Figure 22(a)), let Q := P [a1 , p0s ]∪ Ps ∪ MP [x2 , ps ]. Suppose
there are three disjoint paths from b2 to pw , from a2 to pu , and from b1 to pr in LP −Q. These
paths together with Pu , Pw , and Pr would show that x2 ∈ R(Q), and {x1 , x2 } is feasible
in Γ, a contradiction. Hence no such paths may exist, and there must be a separation as
desired.
The separations guaranteed by Proposition 3.9 may be of several types. Each produces
a slightly different cut in LP . We may assume there are vertices h1 , h2 ∈ V (H ∩ K) with
h1 ∈ MP [b2 , pw ], h2 ∈ MP [x2 , ps ]. If |V (H ∩ K)| = 3, label the third vertex as h3 .
44
We may have {h1 , h2 } a 2-cut in LP , or we may have a 3-cut {h1 , h2 , h3 } as illustrated
in Figure 22(a). (h1 and h3 share a finite face of LP , as do h2 and h3 .) We will refer to this
type of separation as a cutting separation in LP . If h1 and h2 are cofacial with a common
vertex p0 ∈ V (P ), we find a separation (H 0 , K 0 ) in G with V (H 0 ∩ K 0 ) = V (H ∩ K) ∪ {p0 }.
However, V (H ∩ K) may also be a 2-cut in G, with an additional vertex in MP sharing
no finite face of G with the vertices of the 2-cut, as in Figure 23. (In this case h1 and h2
are both incident with the infinite face of LP .) Then either h3 ∈ MP [pr , x2 ) ∪ MP (s, b1 ],
and {h1 , h3 } is a 2-cut in LP , or h3 ∈ [pw , pu ] and {h2 , h3 } is a 2-cut in LP . We will refer
to this type of separation as a non-cutting separation in LP .
Proposition 3.10. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph, and let x1 , x2 ∈ V (G) \
{a1 , b1 , a2 , b2 } such that {x1 , x2 } is infeasible in Γ. Let P ∈ P(Γ,{x1 ,x2 }) be a path from
a1 to x1 , with x2 ∈ V (MP ). Suppose we have four paths Qt , Qw , Qr , and Qs from
0 , q 0 , q 0 ∈ V (P ) to q , q , q , q ∈ V (M ) respectively, such that
qt0 , qw
t w r s
P
r s
1. qt0 6= qs0 , qw 6= qr
0 , x appear in order along P
2. a1 , qt0 , qs0 , qr0 , qw
1
3. qt 6= b2 , qw , qr , x2 , and qs are distinct, and
4. b2 , qt , qw , qr , x2 , qs , b1 appear in order along MP
Then there is a separation (H, K) of LP of order at most three such that {qt , b2 , a2 , b1 } ⊆
V (H) and {qw , qr , x2 , qs } ⊆ V (K).
Proof. Given such a paths (as in Figure 22(b)), let Q := P [a1 , qt0 ] ∪ Qt ∪ MP [qt , qw ] ∪ Qw ∪
0 , x ]. Suppose there are three disjoint paths from b to q , from a to x , and from b
P [qw
1
1
r
2
2
2
to qs in LP − Q. These paths together with Qr and Qs would show that x2 ∈ R(Q), and
{x1 , x2 } is feasible in Γ, a contradiction. Hence no such paths may exist, and there must
be a separation as desired.
As with the separations of Proposition 3.9, those of Proposition 3.10 may take several forms. We assume there are vertices h1 , h2 ∈ V (H ∩ K) with h1 ∈ MP [qt , qw ], h2 ∈
45
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x10011
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Figure 23: x2 cofacial with a1
MP [qs , b1 ]. We then have {h1 , h2 } a 2-cut in LP ; or each of h1 , h2 shares a finite face with
h3 ∈ V (H ∩K); or h3 ∈ MP [x2 , qs ], and {h1 , h3 } is a 2-cut in LP ; or h3 ∈ MP [b2 , qt )∪(qw , qr ]
and {h2 , h3 } is a 2-cut in LP .
Lemma 3.11. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C. Let X ⊆
V (G) \ {a1 , b1 , a2 , b2 } such that X ∩ V (C) = ∅, and X is infeasible in Γ. Let P ∈ P(Γ,X) be
a path from a1 to x1 . Suppose there is a facial cycle D of G with X \ {x1 } ⊆ V (D ∩ MP )
and a1 ∈ V (D). Then (Γ, X) ∈ O3 or (Γ, X) satisfies property (R1).
Proof. Let X 0 = X \ {x1 }, and choose x2 ∈ X 0 to minimize MP [x2 , b1 ]. We assume x1 is not
cofacial with a1 ; otherwise (Γ, X) ∈ O3 . So x1 ∈
/ V (D). Assume by symmetry that D[a1 , x2 ]
is to the right of P (Figure 23). Let S ⊆ D[a1 , x2 ] be a path from a1 to s ∈ MP [x2 , b1 ]. From
Proposition 2.9 there is no edge a1 b1 , and together with (3, {a1 , b1 , a2 , b2 })-connectivity of
G this gives b1 ∈
/ V (D).
Since x1 is not cofacial with a1 , there is some path T to the left of P , from t0 ∈ P (a1 , x1 )
to t ∈ MP [b2 , x2 ). Choose T such that both of t0 , t are cofacial with a1 . Also since x1 is not
cofacial with a1 , there is some path R to the right of P , from r 0 ∈ P (a1 , x1 ) to r ∈ MP [t, x2 ).
Choose R so that both of r 0 , r are cofacial with a1 . Since G is (3, {a1 , b1 , a2 , b2 })-connected,
we see that r 6= t.
If {t, r, a1 } is a 3-cut, this defines a 3-separation of G demonstrating that (Γ, X) ∈ O3 .
Hence we may assume this is not a cut, and there must be some path U from u0 ∈ P (t0 , x1 ]∩
46
P (r 0 , x1 ] to u ∈ MP (t, r). Choose such a path U to minimize MP [t, u]. If t0 ∈ P [r, x1 ) (T
falls at or below R), then by this choice each vertex of MP [t, u] is cofacial with t0 . If
t0 ∈ P (a1 , r 0 ) (T falls above R), then each vertex of MP [t, u] is cofacial with some vertex
v ∈ P [t0 , r 0 ], which in turn is cofacial with a1 .
We may assume D[a1 , x2 ] is disjoint from C[b1 , b2 ]. Otherwise, there is some vertex
v ∈ C(b1 , b2 ) sharing a finite face with a1 . So {a1 , v} is a 2-cut in G, and we have a 4separation (H, K) of G with V (H ∩ K) = {a1 , v, a2 , b2 }, such that {a1 , b2 , a2 , b1 } ⊆ V (H)
and X ⊆ V (K). Hence (Γ, X) satisfies property (R1)
Apply Proposition 3.9 with Ps = S, Pu = U , Pw = T , Pr = R, to find that there must
be a separation (H, K) of LP of order at most three such that {s, b1 , a2 , b2 } ⊆ V (H) and
{x2 , r, u, t} ⊆ V (K). We may assume there are h1 , h2 ∈ V (H ∩ K) with h1 ∈ MP [b2 , t], h2 ∈
MP [x2 , s]. From our choices of T and S, we see that each of h1 , h2 are cofacial with a1 . If
|V (H ∩ K)| = 3, label the third vertex as h3 .
Suppose {h1 , h2 } is a 2-cut in LP , or each of h1 , h2 shares a finite face with h3 . Then
there is a separation (H 0 , K 0 ) of G of order at most 4 with V (H 0 ∩ K 0 ) = V (H ∩ K) ∪ {a1 },
such that {a1 , b2 , a2 , b1 } ⊆ V (H) and X ⊆ V (K). If the separation is of order 4, then (Γ, X)
satisfies property (R1). If it is of order 3, then (Γ, X) ∈ O2 ∩ O3 .
If {h1 , h3 } is a 2-cut in LP with h3 ∈ MP [r, x2 )∪ (s, b1 ], then we have a 3-cut {h1 , h3 , a1 }
in G, and (Γ, X) ∈ O3 . If {h2 , h3 } is a 2-cut in LP with h3 ∈ MP [t, u], then we have some
vertex v ∈ V (P [t0 , r 0 ])∪{t} which is cofacial with each of h3 and a1 . So there is a 4-separation
(H 0 , K 0 ) of G with V (H 0 ∩ K 0 ) = {v, h3 , h2 , a1 }, demonstrating that (Γ, X) satisfies (R1).
Lemma 3.12. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C. Let X ⊆
V (G) \ {a1 , b1 , a2 , b2 } such that X ∩ V (C) = ∅, and X is infeasible in Γ. Let P ∈ P(Γ,X) be
a maximizing path, and assume P is from a1 to x1 . Suppose there is a facial cycle D of G
with X \ {x1 } ⊆ V (D ∩ MP ), and D ∩ P 6= ∅. Then either (Γ, X) ∈ O1 ∪ O2 ∪ O3 ∪ O7 , or
(Γ, X) satisfies property (R1).
Proof. Let X 0 := X \ {x1 }, and choose x2 ∈ X 0 to minimize MP [x2 , b1 ]. By symmetry,
47
assume the face bounded by D is to the right of P . Let dx , da , d2 , d1 be vertices in V (D)
in that clockwise order such that P ∩ D = P [da , dx ] and MP ∩ D = MP [d1 , d2 ] (see Figure
24(a)).
We claim that we may choose D such that x2 6= d1 . To see this, note by our choice of
x2 that if x2 = d1 , we have X = {x1 , x2 }, and there is another facial D 0 with D[d1 , dx ] =
D0 [dx , d1 ]. If x1 ∈ V (D 0 ) then (Γ, X) ∈ O1 , so we assume x1 ∈
/ V (D 0 ), and the face bounded
by D0 is to the right of P .
We assume a1 ∈
/ V (D), so da 6= a1 ; otherwise by Lemma 3.11, (Γ, X) ∈ O3 or
(Γ, X) satisfies property (R1). We also assume x1 ∈
/ V (D); otherwise, (Γ, X) ∈ O1 .
From (3, {a1 , b1 , a2 , b2 })-connectivity of G, together with Lemma 3.2 (each bridge has an
attachment on MP ), there is some path W from w0 ∈ P (dx , x1 ] to w ∈ MP [b2 , d1 ).
Choose such a path W to minimize MP [b2 , w] and, subject to this, to minimize P [dx , w0 ].
By (3, {a1 , b1 , a2 , b2 })-connectivity of G and Lemma 3.3 (limiting paths within (P ∪ LP )bridges), we find that W is not to the right of P . Choose W0 from w00 ∈ P [w0 , x1 ] to w
minimizing P [w00 , x1 ]. (W0 = W is possible.)
We will consider five cases. We first suppose b1 ∈ V (D), and then suppose that there is
some cutvertex of LP in V (D). These situations would prevent any useful routing from a1
to x2 along D[da , x2 ]. We next consider the case that da ∈ C[b2 , a1 ], and then that there is
some cutvertex of LP left of w which is cofacial with da . We close with the case that da is
not cofacial with any vertex of C[b2 , a1 ]. In this case, we will route a path from a1 to x1
along some path left of P , avoiding D.
Case 1
Suppose d2 = b1 .
This case is illustrated in Figure 24. In this case, b1 ∈ V (UP ), so by Lemma 3.4, we
may assume b2 ∈
/ V (UP ). If {da , b1 } is a 2-cut in G, then we have a separation (H, K)
of G with V (H ∩ K) = {da , b1 , a2 , b2 } which demonstrates that (Γ, X) satisfies property
(R1). We may assume this is not the case, so there is some path T from t0 ∈ P [a1 , da ) to
t ∈ MP (b2 , d1 ]. Choose such a path T to minimize P [t0 , da ] and MP [t, d1 ].
Applying Proposition 3.10 with Qt = T , QW = W0 , Qr = D[d1 , dx ], and Qs = D[da , d2 ]
48
(Figure 24(a)), we see that there is a separation (H, K) of LP of order at most three such that
{t, b2 , a2 , b1 } ⊆ V (H) and {w, d1 , d2 } ∪ X 0 ⊆ V (K). We may assume b1 = d2 ∈ V (H ∩ K),
and there is some h1 ∈ V (H ∩ K) with h1 ∈ MP [t, w]. If |V (H ∩ K)| = 3, label the third
vertex h3 .
By our choice of T and W , h1 must be cofacial with some vertex v ∈ P [da , dx ], which is
also cofacial with b1 . If h1 ∈ V (C), and so is a cutvertex of LP , then there is a separation
(H 0 , K 0 ) of G with V (H 0 ∩ K 0 ) = {h1 , v, b1 , a1 }. This separation shows (Γ, X) satisfies (R1).
Similarly, suppose {h1 , b1 } is a 2-cut in LP , or each of h1 , b1 shares a finite face with h3 . Then
there is a separation (H 0 , K 0 ) of G of order at most 4 with V (H 0 ∩ K 0 ) = V (H ∩ K) ∪ {v},
such that {a1 , b2 , a2 , b1 } ⊆ V (H) and X ⊆ V (K). If the separation is of order 4, then (Γ, X)
satisfies property (R1). If it is of order 3, then (Γ, X) ∈ O2 .
If {h1 , h3 } is a 2-cut in LP with h3 ∈ MP [x2 , b1 ], then we have a 3-cut {h1 , h3 , v} in
G, demonstrating (Γ, X) ∈ O2 . We are left with the case that {b1 , h3 } is a 2-cut in LP
with h3 ∈ MP [b2 , t) ∪ (w, d1 ], as illustrated in Figure 24(a). Assume this 2-cut is chosen
with MP [b1 , h3 ] maximal. Suppose h3 ∈ MP [b2 , t). Then there is some vertex v 0 ∈ P [a1 , da ]
which is cofacial with both h3 and b1 , and there is a 3-cut {h1 , h3 , v 0 } in G demonstrating
that (Γ, X) ∈ O2 . Hence we may assume h3 ∈ MP (w, d1 ].
If w is cofacial with h3 , we have a 4-separation (H 0 , K 0 ) of G with V (H 0 ∩ K 0 ) =
{b1 , v, w, h3 }, and (Γ, X) satisfies (R1). Similarly, if x1 shares a finite face with each of v and
h3 , which can occur if w0 = x1 , we have such a separation with V (H 0 ∩ K 0 ) = {b1 , v, x1 , h3 }.
Hence we may assume there is some path R from r 0 ∈ P (v, x1 ) to MP (w, h3 ). Now since G
is (3, {a1 , b1 , a2 , b2 })-connected, {r 0 , r} is not a 2-cut in G, and there is a path U from x1 to
u ∈ MP [t, r).
Apply Proposition 3.10 with Qt = T , QW = U , Qr = R, and Qs = D[da , d2 ] (Figure
24(b)). There is a separation (M, N ) of LP of order at most three such that {t, b2 , a2 , b1 } ⊆
V (M ) and {u, r, b1 } ∪ X 0 ⊆ V (N ). We may assume b1 = d2 ∈ V (M ∩ N ), and there is
some m1 ∈ V (M ∩ N ) with m1 ∈ MP [t, w]. If |V (M ∩ N )| = 3, label the third vertex m3 .
Assume the separation is chosen with MP [b2 , m1 ] minimal.
If m3 ∈ MP [d1 , b1 ] is cofacial with m1 , then m1 is also cofacial with b1 , contradicting
49
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our choice of the maximal 2-cut {b1 , h3 }, so this case does not occur. Also by our choice of
h3 , we do not have m3 ∈ MP [b2 , r] cofacial with b1 .
If m1 is a cutvertex of LP , we must have m3 ∈ MP (w, u] by earlier arguments. Let
Q := P [a1 , t0 ] ∪ T ∪ MP [t, w] ∪ W ∪ P [w0 , x1 ]. Now D ⊆ A(Q), and A(P ) ⊆ A(Q), but
D * A(P ). This contradicts the maximality of P .
Hence we must have either a 2-cut {m1 , b1 }, or each of m1 , b1 sharing a finite face with
m3 in LP . If m1 is cofacial with any vertex of P [da , dx ], we have a separation (M 0 , N 0 )
of G demonstrating that (Γ, X) ∈ O2 or satisfies property (R1). Hence we may assume
m1 ∈ MP (w, u].
Finally, choose x3 ∈ X 0 to minimize MP [d1 , x3 ]. Apply the arguments of Proposition
3.9 to path Ps = P [a1 , dx ] ∪ D[x3 , da ], Pu = U , Pw = W , and Pr = R. (We need a slight,
but obvious, modification to deal with x3 being right of ps . See Figure 24(c).) This gives
some separation (L1 , L2 ) of LP of order at most three such that {b2 , a2 , b1 } ∪ X 0 ⊆ V (L1 )
and {w, u, r, d1 } ⊆ V (L2 ). We may assume b1 ∈ V (L1 ∩ L2 ), and some l1 ∈ V (L1 ∩ L2 ) with
l1 ∈ MP [b2 , w].
By our choices of m1 and h3 , we have either a 2-cut {l1 , b1 }, or each of l1 , b1 sharing a
finite face with l3 in LP . Note that l1 is cofacial either with some vertex v 0 ∈ V (P [da , dx ]) ∪
{a1 }. Hence we have a separation (L01 , L02 ) in G with V (L01 ∩ L02 ) = V (L1 ∩ L2 ) ∪ {v}. If
this separation is of order 3, then (Γ, X) ∈ O2 . If it is of order 4, then (Γ, X) satisfies (R1).
Case 2
Suppose there is a vertex c ∈ V (MP [x2 , d2 ] ∩ C).
If {da , c} is a 2-cut in G, or {da , c, b2 } is a 3-cut in G, then we have a separation (H, K)
of G with V (H ∩ K) = {da , c, a2 , b2 }, {a1 , b1 , a2 , b2 } ⊆ V (H), and X ⊆ V (K), so (Γ, X)
satisfies property (R1). We may assume this is not the case, so there is some path T from
t0 ∈ P [a1 , da ) to t ∈ MP (b2 , d1 ]. Choose such a path T to minimize P [t0 , da ] and MP [t, d1 ].
Applying Proposition 3.10 with Qt = T , Qw = U , Qr = D[d1 , dx ], and Qs = D[da , d2 ],
we see that there is a separation (H, K) of LP of order at most three such that {t, b2 , a2 , b1 } ⊆
V (H) and {u, r, d2 } ⊆ V (K).
Note that d2 ∈
/ A(P ), and by Lemma 3.7 we may assume X 0 ∩ A(P ) 6= ∅. So we
51
may assume there are hi ∈ V (H ∩ K) with h1 ∈ MP [t, u] and one of the following: (i)
h1 ∈ V (C), or (ii) there is h2 ∈ MP [x2 , c] sharing a finite face with h1 in G, or (iii) there
is h3 ∈ V (H ∩ K) with h3 ∈ C[c, a2 ] sharing a finite face with h1 . Note that h1 is cofacial
with some vertex v 0 ∈ P [da , dx ]. In case (i), we have a 4-cut {h1 , a1 , c, v 0 }, and in case (ii)
we have 4-cut {h1 , h2 , c, v 0 }, each showing that (Γ, X) satisfies (R1). In case (iii), 3-cut
{h1 , h2 , v 0 } demonstrates that (Γ, X) ∈ O2 (Γ, X) satisfies (R1).
Case 3
Suppose da ∈ C[b2 , a1 ].
Recall the assumption that a1 ∈
/ V (D), so a1 6= da . Therefore, by 2-connectivity of G,
there is a path S from s0 ∈ P [a1 , da ) to s ∈ MP [d2 , b1 ]. If {da , b1 } is a 2-cut in G, we have
a separation (G1 , G2 ) of G with V (G1 ∩ G2 ) = {da , b1 , a2 , b2 }, {a1 , b1 , a2 , b2 } ⊆ V (G1 ), and
X 0 ⊆ V (G2 ), which demonstrates that (Γ, X) satisfies (R1). So we assume {da , b1 } is not a
2-cut in G, and we may assume s 6= b1 . Choose S so that each vertex of MP [d2 , s] is cofacial
with da .
Define Pw ⊆ C[b2 , a1 ], a path from p0w ∈ V (P ) to pw ∈ V (MP ). Note that x1 ∈
/ V (C),
so we may assume p0w ∈ P [da , x1 ). If {pw , da , d1 } is a 3-cut in G, we have (Γ, X) ∈ O7 .
So we may assume not, and there is a path U from u0 ∈ V (P ) to u ∈ MP (pw , d1 ). By
(3, {a1 , b1 , a2 , b2 })-connectivity of G, we may assume u0 ∈ P (p0w , x1 ] ∩ P (dx , x1 ]. Choose
such a path U so that each vertex of MP [w, u] is cofacial with some vertex of P ∩ C.
Apply Proposition 3.9 with Ps = S, Pu = U , Pw , and Pr = D[d1 , dx ]. (We could use
Ps = D[da , d2 ], but for the possibility that da = dx .) We find a separation (H, K) of LP of
order at most three such that {b2 , a2 , b1 , s} ⊆ V (H) and {pw , u, x2 , d1 } ⊆ V (K). We may
assume there are vertices h1 , h2 ∈ V (H ∩ K) with h1 ∈ MP [b2 , pw ] and h2 ∈ MP [x2 , s]. If
|V (H ∩ K)| = 3, label the third vertex as h3 .
By 2-connectivity of G, there is no cutvertex of LP between b2 and pw . Suppose h2 shares
a finite face with h1 in G, or h3 shares a finite face with each of h1 , h2 , so (H, K) is a cutting
separation. Then we have a separation (H 0 , K 0 ) of G with V (H 0 ∩ K 0 ) = V (H ∩ K) ∪ {da }.
If this is of order 3, (Γ, X) ∈ O2 . If it is of order 4, (Γ, X) satisfies (R1).
Now suppose h3 ∈ MP [d1 , x2 ) shares a finite face with h1 . Then {h1 , h3 , da } is a 3-cut
52
in G, and (Γ, X) ∈ O7 . If h3 ∈ MP (x2 , b1 ] shares a finite face with h1 , then there is some
vertex v 0 ∈ P [a1 , da ] cofacial with each of h1 and h3 . Hence {h1 , h3 , v 0 } is a 3-cut in G, and
(Γ, X) ∈ O2 .
Finally, if h3 ∈ MP (pw , u] shares a finite face with h2 , then by our choice of U , h3 is
cofacial with some vertex p ∈ V (P ∩ C). Then we have a 4-separation (H 0 , K 0 ) of G with
V (H 0 ∩ K 0 ) = {h2 , h3 , da , p}, and (Γ, X) satisfies (R1).
Case 4
Suppose da ∈
/ C[b2 , a1 ] is cofacial with some b ∈ MP [b2 , w] ∩ C[b1 , b2 ].
Assume b is chosen to minimize MP [b, w]. Recall the assumption that a1 ∈
/ V (D). If
{da , b} is a 2-cut in G, or {da , b, b1 } is a 3-cut in G, we have a separation (G1 , G2 ) of G
with V (G1 ∩ G2 ) = {da , b, b1 , a2 }, {a1 , b1 , a2 , b2 } ⊆ V (G1 ), and X ⊆ V (G2 ). Hence (Γ, X)
satisfies (R1). So we may assume no such cut exists, and da ∈
/ V (C). Furthermore, there are
paths S0 from s00 ∈ P [a1 , da ) to s0 ∈ MP [d2 , b1 ), and T0 from t00 ∈ P (s00 , da ] to t0 ∈ MP [b2 , b],
where t0 = b only if b = b2 ; otherwise {b, a1 , b1 , a2 } is a 4-cut in G, producing a separation
of G which shows that (Γ, X) satisfies property (R1). Choose such a pair S0 and T0 so
that each vertex of MP [t0 , w] is cofacial with some vertex of P [da , dx ], and each vertex of
MP [d2 , s0 ] is cofacial with some vertex of P [t00 , da ].
Suppose there is no path from P (da , w00 ) to MP [b, w). Then there is a facial cycle of G
containing V (MP [t, w]) ∪ V (P [t0 , da ]). Let Q := P [a1 , s00 ] ∪ S0 ∪ MP [x2 , s0 ]. If there are
disjoint paths from b2 to t0 , from a2 to w, and from b1 to d1 in LP , each disjoint from
MP [x2 , s0 ], then we find x1 ∈ R(Q), and X is feasible in Γ, a contradiction. Hence no such
disjoint paths may exist. So we have either (i) a cutvertex z of LP with z ∈ MP [x2 , s0 ], or
(ii) a 2-separation (H, K) of LP with {a2 , b1 , s0 } ⊆ H and {w, d1 , x2 } ⊆ V (K).
If z ∈ MP [x2 , s0 ] is a cutvertex of LP , note that z is cofacial with some vertex v ∈
P [t00 , da ], which is in turn cofacial with b. So there is a 4-separation (M, N ) of G with
V (M ∩ N ) = {b, a1 , z, v}, {a1 , b1 , a2 , b2 } ⊆ V (M ), and X ⊆ V (N ), and (Γ, X) satisfies
property (R1).
So we assume there is a 2-separation (H, K) of LP with {a2 , b1 , s0 } ⊆ H and {w, d1 , x2 } ⊆
V (K). Letting V (H ∩ K) = {h1 , h2 }, we may assume h2 ∈ MP [x2 , s0 ]. Here h2 is cofacial
53
with some vertex v ∈ P [t00 , da ], which is in turn cofacial with b. If h1 ∈ C[a2 , b), then we
have a 4-separation (H 0 , K 0 ) of G with V (H 0 ∩ K 0 ) = V (H ∩ K) ∪ {b, v}, demonstrating that
(Γ, X) satisfies (R1). If h1 ∈ MP [b, w], then h1 is cofacial with v, and we have a 3-separation
(H 0 , K 0 ) of G with V (H 0 ∩ K 0 ) = V (H ∩ K) ∪ {v}, demonstrating that (Γ, X) ∈ O3 .
Now suppose there is some path T from t0 ∈ P (da , w00 ) to t ∈ MP [b, w). Let T be chosen
to minimize P [da , t0 ] and MP [b, t], and let S be a path from s0 ∈ P [a1 , da ] to s ∈ MP [d1 , b1 ),
chosen to minimize P [s0 , da ] and MP [d1 , s]. Now each vertex of MP [x2 , s] is cofacial with
da , as is each vertex of MP [b, t].
Apply Proposition 3.9 with Ps = S, Pu = W0 , Pw = T , and Pr = D[d1 , dx ]. We
find a separation (H, K) of LP of order at most three such that {b2 , a2 , b1 , s} ⊆ V (H)
and {t, w, d1 , x2 } ⊆ V (K). We may assume there are vertices h1 , h2 ∈ V (H ∩ K) with
h1 ∈ MP [b2 , t], h2 ∈ MP [x2 , s]. If |V (H ∩ K)| = 3, label the third vertex as h3 . We
now have one of (i) h2 is a cutvertex in LP , (ii) (H, K) is a cutting separationin LP , (iii)
h3 ∈ MP [b, w] shares finite face of LP with h2 , or (iv) h3 ∈ MP [d1 , x2 ] shares a finite face
of LP with h1 .
If h2 is a cutvertex of LP , then there is a 4-separation (H 0 , K 0 ) of G with V (H 0 ∩ K 0 ) =
{b, a1 , h2 , da }, {a1 , b1 , a2 , b2 } ⊆ V (H 0 ), and X ⊆ V (K 0 ), and (Γ, X) satisfies property (R1).
If (H, K) is a cutting separation, then there is a separation (H 0 , K 0 ) of G with V (H 0 ∩
K 0 ) = V (H ∩ K) ∪ {da }. If this is of order three, then (Γ, X) ∈ O2 . If of order 4, then
(Γ, X) satisfies (R1).
Suppose h3 ∈ MP [b, w] shares finite face with h2 . If h3 shares a finite face with da , we
have a 3-cut {h3 , h2 , da } in G demonstrating (Γ, X) ∈ O2 . Otherwise, h3 is cofacial with
some p ∈ P (da , dx ], and we have a 4-cut {h2 , h3 , da , p}. An application of Lemma 2.11 gives
(Γ, X) ∈ O2 , or (Γ, X) satisfies property (R1).
We now assume h3 ∈ MP [d1 , x2 ) ∪ MP (s, b1 ] shares a finite face of LP with h1 . We claim
that we may assume t0 ∈ MP [b2 , h1 ). To see this, first note that t0 ∈
/ MP [b2 , h1 ) if and only
if t0 = h1 = b. Recalling that t0 = b only if b = b2 , this occurs only if h1 = b2 . Suppose first
that h3 ∈ MP [d1 , x2 ). So there is a 3-cut {h1 , h3 , da } in G. If h1 = b2 , then (Γ, X) ∈ O7 .
54
Now suppose h3 ∈ MP (s, b1 ]. If h1 = b2 , then we have some vertex v 0 ∈ P [a1 , da ] which is
cofacial with each of h3 and b2 , so {h1 , h3 , v 0 } is a 3-cut in G, and (Γ, X) ∈ O2 .
Let Q := P [a1 , s00 ] ∪ S0 ∪ MP [x2 , s0 ]. If there are disjoint paths from b2 to t0 , from a2
to w, and from b1 to d1 in LP , each disjoint from MP [x2 , s0 ], then we find x1 ∈ R(Q), and
X is feasible in Γ, a contradiction. Hence no such disjoint paths may exist. So we have
either (i) a cutvertex z of LP with z ∈ MP [x2 , s0 ], or (ii) a 2-separation (M, N ) of LP with
{a2 , b1 , s0 } ⊆ M and {w, d1 , x2 } ⊆ V (N ).
If z ∈ MP [x2 , s0 ] is a cutvertex of LP , note that z is cofacial with some vertex v ∈
P [t00 , da ], which is in turn cofacial with b. So there is a 4-separation (M 0 , N 0 ) of G with
V (M 0 ∩ N 0 ) = {b, a1 , z, v}, {a1 , b1 , a2 , b2 } ⊆ V (M 0 ), and X ⊆ V (N 0 ), and (Γ, X) satisfies
property (R1).
So we assume there is a 2-separation (M, N ) of LP with {a2 , b1 , s0 } ⊆ M and {w, d1 , x2 } ⊆
V (N ). Letting V (M ∩N ) = {m1 , m2 }, we may assume m2 ∈ MP [x2 , s0 ]. Here m2 is cofacial
with some vertex v ∈ P [t00 , da ], which is in turn cofacial with b. If m1 ∈ C[a2 , b), then we
have a 4-separation (M 0 , N 0 ) of G with V (M 0 ∩ N 0 ) = V (M ∩ N ) ∪ {b, v}, demonstrating
that (Γ, X) satisfies (R1). Finally, if m1 ∈ MP [b, w], then either m1 is cofacial with v, or
h1 , m2 is a 2-cut in LP , with each of h1 , m2 cofacial with v. Hence we have a 3-separation
(M 0 , N 0 ) of G demonstrating that (Γ, X) ∈ O3 .
Case 5
Suppose none of Cases 1-4 hold.
Here da is not cofacial with any vertex of MP [b2 , w] ∩ C[b1 , b2 ], and there is some path
T from t0 ∈ P [a1 , da ) to t ∈ MP (b2 , d1 ]. Choose such a path T to minimize P [t0 , da ] and
MP [t, d1 ]. Choose a path Dx from d0x ∈ P [dx , w00 ) to d01 ∈ MP (w, d1 ] to minimize P [d0x , w00 ]
and MP [w, d01 ]. (Dx = D[d1 , dx ] is possible.)
Applying Proposition 3.10 with Qt = T , Qw = W0 , Qr = Dx , and Qs = D[da , d2 ],
we find that there must be a separation (M, N ) of LP of order at most three such that
{t, b2 , a2 , b1 } ⊆ V (M ) and {w, d01 , d2 } ∪ X 0 ⊆ V (N ). We may assume there are vertices
m1 , m2 ∈ V (M ∩ N ) with m1 ∈ MP [t, w], m2 ∈ MP [d2 , b1 ]. If |V (M ∩ N )| = 3, label the
third vertex as m3 . Notice from our choice of T and W0 that m1 must be cofacial with
55
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Figure 25: Separations when s0 not cofacial with b2
some v ∈ P [da , dx ] (Figure 25(a)). We have one of the following situations:
(i) m2 shares a finite face of LP with m1 in G, or
(ii) m3 shares a finite face LP with each of m1 , m2 , or
(iii) m3 ∈ MP [x2 , d2 ] shares a finite face LP with m1 , or
(iv) m3 ∈ MP [b2 , t) ∪ MP (w, d01 ] shares a finite face LP with m2 .
In case (iii), we have each of m1 , m3 cofacial with v, so {m1 , m3 , v} is a 3-cut demonstrating that (Γ, X) ∈ O2 . We may assume, then, that case (iii) does not occur.
(5.1)
Suppose (i) or (ii) holds, so (M, N ) is a cutting separation.
Assume S and R are chosen to minimize P [a1 , s0 ], MP [s, b1 ], P [a1 , r 0 ], and MP [b2 , r].
Suppose m1 and m2 are both cofacial with some v 0 ∈ P [t0 , s0 ]. Then we have a separation
(M 0 , N 0 ) of G with V (M 0 ∩ N 0 ) = V (M ∩ N ) ∪ {v 0 }. In case (i), (M 0 , N 0 ) is a 3-separation,
and (Γ, X) ∈ O2 . In case (ii), (M 0 , N 0 ) is a 4-separation, and by Lemma 2.11, (Γ, X) ∈ O2
or (Γ, X) satisfies (R1).
Hence we may assume that any vertex v 0 ∈ P [t0 , s0 ] which is cofacial with m1 is not
also cofacial with m2 . So there are paths S from s0 ∈ P [a1 , da ] to s ∈ MP [d2 , m2 ) and
R from r 0 ∈ P (s0 , w0 ] to r ∈ MP [t, m1 ). (If there is a path “between” T and W , we get
56
S = D[da , d2 ]. If not, we get some S above T , and R = T .) From our choice of W , we may
assume r 0 ∈ P (s0 , dx ], and R is thus disjoint from W .
Apply Proposition 3.9 with Ps = S, Pu = W , Pw = R, and Pr = D[d1 , dx ], to find that
there is a separation (H, K) of LP of order at most three such that {b2 , a2 , b1 , s} ⊆ V (H)
and {r, w, d1 } ∪ X 0 ⊆ V (K). We may assume there are vertices h1 , h2 ∈ V (H ∩ K) with
h1 ∈ MP [b2 , r] and h2 ∈ MP [x2 , s]. If |V (H ∩ K)| = 3, label the third vertex as h3 (Figure
25(b)). We have one of the following situations:
(a) h2 shares a finite face of LP with h1 in G, or
(b) h3 shares a finite face of LP with each of h1 , h2 , or
(c) h3 ∈ MP [d1 , x2 ) ∪ MP (s, b1 ] shares a finite face of LP with h1 , or
(d) h3 ∈ MP (r, w] shares a finite face of LP with h2 .
In case (d), h3 is cofacial with some v 0 ∈ P [da , dx ], which is also cofacial with h2 . Hence
there is a 3-cut {h2 , h3 , v 0 }, demonstrating that (Γ, X) ∈ O2 . We may assume, then, that
case (d) does not occur.
Suppose h3 ∈ MP (m2 , b1 ] shares a finite face of LP with h1 . By our choice of S and R,
h1 and h3 are cofacial with a common vertex p ∈ P [a1 , r 0 ]. So we have a separation (M 0 , N 0 )
of G with V (M 0 ∩ N 0 ) = {h1 , h3 , p}, {a1 , b2 , a2 , b1 } ⊆ V (M 0 ) and {x1 , x2 } ⊆ V (N 0 ), and
(Γ, X) ∈ O2 . So we may assume that if (c) occurs, h3 ∈ MP [d1 , m2 ].
We now have the vertices b2 , h1 , r, m1 , w, d01 , d1 , x2 , h2 , s, m2 , b1 in that order along MP .
(See Figure 26.) By planarity, either {h1 , m2 } is a 2-cut in LP , or there is some m ∈
{h3 , m3 } such that {h1 , m2 , m} is a 3-cut in LP . By our choice of S and R, h1 and m2
are cofacial with a common vertex p ∈ P [a1 , r 0 ]. So we have a separation (M 0 , N 0 ) of
G with V (M 0 ∩ N 0 ) = {h1 , m2 , m, p}, {a1 , b2 , a2 , b1 } ⊆ V (M 0 ) and {x1 , x2 } ⊆ V (N 0 ). If
|V (M 0 ∩ N 0 )| = 3, then (Γ, X) ∈ O2 . If |V (M 0 ∩ N 0 )| = 4, note that there are disjoint paths
from h1 to x1 (along W ), from p to m (along D), and from m2 to x2 in N 0 . By Lemma
2.11, either (Γ, X) ∈ O2 , or (Γ, X) satisfies property (R1).
57
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Figure 26: Combinations of separations, Case (5.1)
(5.2)
Suppose (iv) holds: m3 ∈ MP [b2 , t) ∪ MP (w, d01 ] shares a finite face LP with m2 .
We assume (M, N ) is chosen to maximize MP [m3 , m2 ].
Choose b01 ∈ MP [m2 , b1 ] ∩ C[b1 , b2 ] to minimize MP [m2 , b01 ]. So either b01 is a cutvertex
of LP separating a2 from b1 , or b1 ∈ A(P ) and b01 = b1 .
Let S be a path from s0 ∈ P [a1 , x1 ] to s ∈ MP [d2 , b01 ) (D[da , d2 ] is one such path), chosen
to minimize P [a1 , s0 ] and MP [s, b01 ], so each vertex of MP [s, b01 ] is cofacial with s0 . Let T0
be a path from t00 ∈ P (s0 , w0 ] to t0 ∈ MP [b2 , w], chosen to minimize P [s0 , t00 ] and MP [b2 , t0 ]
(T0 = W is possible). So each vertex of MP [b2 , t0 ] is cofacial with some vertex v ∈ P [a1 , s0 ],
and by an application of Lemma 3.3 (controlling paths within (P ∪ LP )-bridges), v is in
turn cofacial with b01 .
Claim 1. We may assume T0 is left of P .
Proof of Claim 1 Since W is not right of P , we have either T0 left of P or t00 = x1 .
Suppose t00 = x1 . Then x1 is cofacial with each vertex of P [s0 , x1 ], and x1 is cofacial with
m3 . Letting v ∈ P [s0 , x1 ] be some vertex cofacial with m2 , we have a 4-separation (G1 , G2 )
of G with V (G1 ∩ G2 ) = {m2 , m3 , x1 , v}, {a1 , b1 , a2 , b2 } ⊆ V (G1 ), X 0 ⊆ V (G2 ). Note that
D[da , dx ] provides a path from v to m3 in G2 , disjoint from MP [xn , m2 ]. By Lemma 2.11,
(Γ, X) ∈ O2 or (Γ, X) satisfies property (R1). Hence we may assume t00 6= x1 , and this
proves Claim 1.
58
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Figure 27: Case (5.2)
Choose a path R from r 0 ∈ P (s0 , x1 ] to r ∈ MP [w, d1 ] to minimize MP [r, d1 ], and subject
to this to minimize P [s0 , r 0 ].
Claim 2. We may assume R is right of P .
Proof of Claim 2 If R is left of P , we find that one of {r 0 , r} or {s0 , r} is a 2-cut in G,
contradicting (3, {a1 , b1 , a2 , b2 })-connectivity of G. So R is not left of P . Suppose r 0 = x1 .
Then x1 shares a facial cycle F to the right of P with s0 , and hence s0 = da = dx . Moreover,
there is vertex p ∈ V (P [s0 , x1 ]) ∩ V (F ) which is cofacial with m3 . So there is a 4-separation
(G1 , G2 ) of G with V (G1 ∩ G2 ) = {m3 , m2 , s0 , p}, {a1 , b1 , a2 , b2 } ⊆ V (G1 ), and X ⊆ V (G2 ).
By Lemma 2.11, (Γ, X) ∈ O2 or (Γ, X) satisfies property (R1). Hence we may assume
r 0 6= x1 . This proves Claim 2.
Now by Claim 2 and (3, {a1 , b1 , a2 , b2 })-connectivity of G, there is some path U from
u0 ∈ P (r 0 , x1 ] ∩ P (t00 , x1 ] to u ∈ MP [t0 , r].
Claim 3. We may assume U is chosen with u ∈ MP (t0 , m3 ), u 6= r.
Proof of Claim 3
Suppose r 0 and m3 share a finite face left of P . By our choice of R
and S, we have each of r 0 and m2 cofacial with s0 , and there is a 4-separation (G1 , G2 ) of G
with V (G1 ∩ G2 ) = {m3 , m2 , r 0 , s0 }, {a1 , b1 , a2 , b2 } ⊆ V (G1 ), and X ⊆ V (G2 ). By Lemma
59
2.11, (Γ, X) ∈ O2 or (Γ, X) satisfies property (R1). Hence we may assume r 0 and m3 do
not share a finite face left of P , so u ∈ MP [t0 , m3 ).
By our choice of T0 , together with Lemma 3.3, we have some vertex v ∈ P [a1 , t00 ] which
is cofacial with both of t0 and m2 . If m3 is cofacial with t0 , then there is a 4-separation
(G1 , G2 ) of G with V (G1 ∩ G2 ) = {m3 , m2 , v, t0 }, {a1 , b1 , a2 , b2 } ⊆ V (G1 ), and X ⊆ V (G2 ).
By Lemma 2.11, (Γ, X) ∈ O2 or (Γ, X) satisfies property (R1). Hence we may assume t0 is
not cofacial with m3 , and thus u ∈ MP (t0 , m3 ).
Now suppose we must choose u = r, so r ∈ MP (t0 , m3 ). For any vertex v ∈ P [s0 , r 0 ],
{v, r} is not a 2-cut in G. Hence there must be some path U0 from x1 to u0 ∈ V (P [a1 , s0 )) ∪
{t0 }. By Lemma 3.3, we find u0 = t0 . Together with our choice of T0 , this gives T0 = W ,
U0 = W0 . But then w00 = x1 , r 0 ∈ P [dx , w00 ), and r ∈ MP (w, m3 ). This contradicts the
choice of Dx and {m2 , m3 }. Hence we may assume u 6= r, which completes the proof of
Claim 3.
Apply Proposition 3.9 with Ps = S, Pu = U , Pw = T0 , and Pr = R, to find that there
is a separation (H, K) of LP of order at most three such that {b2 , a2 , b1 , s} ⊆ V (H) and
{t0 , u, r, x2 } ∪ X 0 ⊆ V (K). We may assume there are vertices h1 , h2 ∈ V (H ∩ K) with
h1 ∈ MP [b2 , t0 ] and h2 ∈ MP [x2 , s]. If |V (H ∩ K)| = 3, label the third vertex as h3 . (See
Figure 28.) We have one of the following situations:
(a) h2 shares a finite face of LP with h1 in G, or
(b) h3 shares a finite face of LP with each of h1 , h2 , or
(c) h3 ∈ MP [r, x2 ) ∪ MP (s, b01 ] shares a finite face of LP with h1 , or
(d) h3 ∈ MP (t0 , u] shares a finite face of LP with h2 .
In case (a), we see by planarity that {h1 , m2 } is a 2-cut in G, contradicting our choice
of {m2 , m3 }. Similarly in case (d), {h3 , m2 } is a 2-cut in G, contradicting our choice of
{m2 , m3 }. Hence we may assume cases (a) and (d) do not occur.
Suppose case (c) occurs, with h3 ∈ MP (s, b01 ] sharing a finite face of LP with h1 . If h3
is right of m2 , we have a 2-cut {h1 , h3 } in G. If h3 is left of m2 , then {h1 , m2 } is a 2-cut in
60
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Figure 28: Combinations of separations, Case (5.2)
G. In each case, the 2-cut contradicts our choice of {m2 , m3 }. So we may assume that of
(c) occurs, we have h3 ∈ MP [r, x2 ).
We now have the vertices b2 , h1 , t0 , u, d1 , x2 , h2 , s, m2 , b01 in that order along MP . (See
Figure 28.) By planarity, either {h1 , m2 } is a 2-cut in LP , or {h1 , m2 , h3 } is a 3-cut in LP .
By our choice of {m2 , m3 }, we must have {h1 , m2 , h3 } a 3-cut in LP .
By our choice of paths S and T0 , we have some vertex v 0 ∈ P [a1 , t00 ] which is cofacial
with both h1 and m2 . Hence there is a 4-separation (H 0 , K 0 ) of G with V (H 0 ∩ K 0 ) =
{h1 , h3 , m2 , v 0 }, {a1 , b1 , a2 , b2 } ⊆ V (H 0 ∩ K 0 ), and X ⊆ V (K). By an application of Lemma
2.11, either (Γ, X) ∈ O2 , or (Γ, X) satisfies property (R1).
Corollary 3.13. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C. Let {x1 , x2 } ⊆
V (G) \ {a1 , b1 , a2 , b2 } such that {x1 , x2 } ∩ V (C) = ∅, and {x1 , x2 } is infeasible in Γ. Let
P ∈ P(Γ,X) be a maximizing path, and assume P is from a1 to x1 . Then (Γ, {x1 , x2 }) ∈
Θ1 ∪ Θ2 ∪ Θ3 , or satisfies property (R1).
3.5
Separating X from a
Here we consider the special case where X is separated from a2 by a 2-cut in LP .
Lemma 3.14. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C. Let X ⊆ V (G)\
{a1 , b1 , a2 , b2 }, with |X| ≥ 2, X ∩ V (C) = ∅, and X infeasible in Γ. Let P ∈ P(Γ,X) be a
maximizing path, and assume P is from a1 to x1 . Assume further that there is some x2 ∈ X
61
such that x2 ∈
/ V (MP ). If there is some 2-separation (H, K) of LP with X \ {x1 } ⊆ V (K)
and {b1 , a2 , b2 } ⊆ H, then either (Γ, X) satisfies property (R1) or (Γ, X) ∈ O1 ∪O2 ∪O3 ∪O5 .
The proof of Lemma 3.14 is given in the next several subsections. In Section 3.5.1, we
define several paths from P to LP which will exist in any counterexample to Lemma 3.14.
(Recall that when we refer to a path from P to LP , we mean a path which is internally
disjoint from P ∪ LP .) One of these is the path T from t0 ∈ V (P ) to t ∈ V (MP ). We then
show that the path T is not to the right of P in any counterexample to the lemma. In
Section 3.5.2, we address the case that t0 = x1 , and in Section 3.5.3 we address the case
that T is to the left of P .
Paths and vertex labels defined within the proof of any single claim will apply only
within that proof. To limit the notation required, they may be redefined in subsequent
claims. Labels defined outside any claim or case will apply to all subsequent arguments in
the section, unless clearly noted otherwise.
Note that since X ∩ V (C) = ∅, if (Γ, X) is reducible then (Γ, X) must satisfy property
(R1).
3.5.1
Defining common paths
Let X 0 = X \ {x1 }. Assume (H, K) is chosen with K maximal, and let Z = V (H ∩ K).
Then Z ⊆ V (MP ). Label Z = {z1 , z2 } such that z1 lies to the left of z2 . Since x2 ∈
V (K) \ V (MP ) 6= ∅, Z is a 2-cut in LP . Let B be the Z-bridge of LP containing X 0 .
We may assume X 0 ⊆ A(P ). Otherwise, we apply Lemma 3.7 to find (Γ, X) satisfies
property (R1) or (Γ, X) ∈ O1 ∪ O2 ∪ O3 ∪ O5 .
Define paths S and T :
Since G is (3, {a1 , b1 , a2 , b2 })-connected, Z is not a 2-cut in G, so
there must be some path S in G from s0 ∈ V (P ) to s ∈ MP (z1 , z2 ). Choose S to minimize
P [a1 , s0 ]. We may assume by symmetry that S is to the right of P , and we further choose
S to minimize MP [s, z2 ]. If there are multiple paths from s0 to MP (z1 , z2 ), label these as
S1 , S2 , . . . , Sk = S with attachments s1 , s2 , . . . , sk in order from z1 to z2 . (See Figure 29(a).)
Note that if there is a separation (M, N ) of G with V (M ∩N ) = {s0 , z1 , z2 }, {a1 , b1 , a2 , b2 } ⊆
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(a) S1 , S2 , . . . , Sk
(b) Path T right of P
Figure 29: Initial path definitions
V (M ) and X ⊆ V (N ), we have (Γ, X) ∈ O2 . Hence we may assume there is no such separation, so by the choice of S there is some path T from P (s0 , x1 ] to P [a1 , s0 ) ∪ MP [b2 , z1 ).
By Lemma 3.2 (every (P ∪ LP )-bridge has an attachment in LP ), there is a path T from
t0 ∈ P (s0 , x1 ] to t ∈ MP [b2 , z1 ). Choose such a T to minimize MP [b2 , t], and subject to this
minimizing P [s0 , t0 ].
Claim 1. We may assume either t0 = x1 or T is to the left of P .
Proof. Suppose not, so t0 6= x1 and T is to the right of P (Figure 29(b)). From our choice
of T to minimize MP [b2 , t], we find that there is no path from P (s0 , x1 ] to MP [b2 , t). From
planarity, there is no path from P (t0 , x1 ] to MP (t, b1 ].
Since G is (3, {a1 , b1 , a2 , b2 })-connected, the set {t0 , t} is not a 2-cut in G, so there must
be some path W from w0 ∈ P [a1 , t0 ) to w ∈ P (t0 , x1 ] internally disjoint from P ∪ LP . Now
from Lemma 3.2 (every P ∪ LP -bridge has an attachment in LP ), we may assume W lies
entirely in some (P ∪ LP )-bridge of G which has attachment t on LP . From our choice of
T to minimize P [s0 , t0 ], we must have w0 ∈ P [a1 , s0 ].
Now by Lemma 3.3 (limiting paths within (P ∪ LP )-bridges), there can be no path
from P (w0 , t0 ] to MP (t, b1 ]. Hence w0 = s0 , and {s0 , t} is a 2-cut in G, violating the
(3, {a1 , b1 , a2 , b2 })-connectivity of G.
63
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Figure 30: Defining path R
Define paths Us and Qx :
Let Us be a path from u0s ∈ P [s0 , x1 ] to us ∈ MP (z1 , s], chosen
to minimize MP [z1 , us ], and subject to this, to minimize P [u0s , x1 ]. Note that if u0s = s0 , we
have Us = S1 .
Now from (3, {a1 , b1 , a2 , b2 })-connectivity of G, there can be no 2-cut of LP contained
in MP [z1 , us ] and separating x2 from z2 in B. Since we assume x2 ∈
/ V (MP ), there are
disjoint paths Qb = MP [z1 , us ] and Qx from z2 to x2 in B.
Define path R: Note from our choices of S and T that t is cofacial with s0 , and s0 is
cofacial with z2 . We also have z2 cofacial with z1 .
Suppose t is cofacial with z1 . Then there is 4-separation (M, N ) of G with V (M ∩ N ) =
{z1 , z2 , s0 , t}, X ⊆ V (N ), and {a1 , b1 , a2 , b2 } ⊆ V (M ). If P [t0 , x1 ] ⊆ V (N ), we have the
following disjoint paths in N : S1 ∪ MP [z1 , s1 ] from s0 to z1 , T ∪ P [t0 , x1 ] from t to x1 , and
Qx from z2 to x2 . Applying Lemma 2.11 (4-separations in G), we find that (Γ, X) ∈ O2 , or
(Γ, X) is reducible. So we may assume that either t is not cofacial with z1 , or for any such
separation x1 ∈
/ V (N ). Therefore there is some path R from r 0 ∈ P [s0 , x1 ] to r ∈ MP (t, z1 ).
We claim that we may choose R such that r 0 6= s0 . (See Figure 30.) To see this, suppose
not. Then any path from P (s0 , x1 ] to MP [t, s1 ) has endpoint t, and {s0 , t} is a 2-cut in G
violating (3, {a1 , b1 , a2 , b2 })-connectivity.
64
3.5.2
The case t0 = x1 .
Claim 2. We may assume R falls to the right of P
Proof. From our choice of T , we see that P [s0 , x1 ] is contained in the boundary of a face to
the left of P . Suppose x1 is cofacial with z1 . Then {z1 , z2 , s0 , x1 } is a 4-cut in G. Recalling
the existence of the paths Qb and Qx , from us to z1 and x2 to z2 respectively, we apply
Lemma 2.11 (4-separations in G) to see that either (Γ, X) is reducible, or (Γ, X) ∈ O2 .
Hence we may assume x1 is not cofacial with z1 .
Similarly, if s0 is cofacial with z1 , then {z1 , z2 , s0 } is a 3-cut in G, and {z1 , z2 , s0 , x1 } is
a 4-cut in G. An application of Lemma 2.11 shows (Γ, X) is reducible, or (Γ, X) ∈ O2 . So
we may assume s0 is not cofacial with z1 .
We have shown that neither x1 nor s0 is cofacial with z1 . Together with our choice of R
and the planarity of G, this proves the claim.
Define path R0 :
Furthermore, since s0 is not cofacial with z1 , there is some path R0 from
r00 ∈ P (s0 , r 0 ] to r0 ∈ MP (z1 , s]. Choose such a path R0 to minimize P [s0 , r00 ] and MP [r0 , s].
Define path T0 :
Noting that x1 ∈
/ V (C), we see that there is a path T0 from t00 ∈ P [a1 , s0 ]
to t0 ∈ MP [b2 , t]. Choose such a path with t00 6= s0 if possible, and then to minimize P [t00 , s0 ]
and MP [t0 , t]. Observe that each vertex of T0 ∪ MP [t0 , t] ∪ T ∪ P [t00 , x1 ] is cofacial with
s0 . (This will in fact be a face boundary unless there is some path from s0 to MP (t0 , t).)
Moreover, if t00 = s0 , we have P [a1 , s0 ] ⊆ V (C).
Claim 3. We may assume C[a2 , b2 ] ∩ MP [t0 , t] = ∅.
Proof. To see this, suppose there is some vertex zt ∈ C[a2 , b2 ] ∩ MP [t0 , t]. (The example
zt = t0 = b2 is illustrated in Figure 31.) If t0 = b2 , we recall Proposition 2.9, a1 b2 is not an
edge of G, so t00 6= a1 . If {zt , s0 } is a 2-cut separating {a1 , b2 } from {b1 , a2 }∪X in G, then we
have a 4-separation (M, N ) of G with V (M ∩ N ) = {zt , s0 , b1 , a2 }, {a1 , b1 , a2 , b2 } ⊆ V (M ),
and X ⊆ V (N ), and (Γ, X) satisfies property (R1). So, by our choice of S, there must
be some path S0 from s00 ∈ P [a1 , t00 ) to s0 ∈ MP [z2 , b1 ]. Choose such an S0 to minimize
P [s00 , t00 ] and MP [z2 , s0 ], so each of s00 , s0 is cofacial with t00 .
65
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Figure 31: Where t0 = b2
For any vertex v ∈ MP [z2 , s0 ], the set {v, zt , s0 } is not a 3-cut separating {a1 , b1 } from
{a2 } ∪ X in G, or G would have a 4-separation (H, K) with V (H ∩ K) = {v, zt , s0 , a2 }
demonstrating that (Γ, X) satisfies (R1). So we may assume s0 ∈
/ C[b1 , a2 ], and s0 ∈ A(P ).
Define Q := P [a1 , s00 ] ∪ S0 ∪ MP [z2 , s0 ] ∪ Qx , a path from a1 to x2 .
We first establish that Q ∈ P(Γ,X) . Suppose not. Then a2 is a cutvertex of LQ , and a2
is cofacial with some vertex q ∈ V (Q). We may assume q ∈ MP [z2 , s0 ] (because of the 2-cut
Z in LP ). But then we have a 4-separation (M, N ) of G with V (M ∩ N ) = {a2 , q, t00 , zt },
and we find (Γ, X) satisfies property (R1). Hence we may assume a2 is not a cutvertex of
LQ , and Q ∈ P(Γ,X) .
Note that (T0 ∪ Us ∪ Qb ) ⊆ LQ . If there are disjoint paths in LP from a2 to r and from b1
to z1 respectively, each disjoint from MP [z2 , s0 ] and b2 , then we find that x1 ∈ R(Q). Hence
no such paths may exist. By our choice of Z as a maximal 2-cut, we see that there must be
some 3-cut {zt , h1 , h2 } in LP separating {a2 , b1 , s0 } from {r, z1 , z2 }, with h1 ∈ MP [z2 , s0 ].
Now h1 is cofacial with some vertex v ∈ P [s0 , x1 ], and v is also cofacial with zt . So G has
a separation (H 0 , K 0 ) of order 4 with V (H 0 ∩ K 0 ) = {zt , h1 , h2 , v}, {a1 , b1 , a2 , b2 } ⊆ V (H 0 ),
and X ⊆ V (K 0 ). Note that we have disjoint paths from zt to x1 , from v to h2 , and from h1
to x2 in K 0 . Applying Lemma 2.11, we find that (Γ, X) ∈ O2 or (Γ, X) satisfies property
(R1).
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a2
Figure 32: Where path Ox does not exist
Define path R1 : Let R1 be a path from r10 ∈ P (t00 , s0 ] to r1 ∈ MP [s, b1 ], if such a path
exists, chosen to minimize P [t00 , r10 ] and MP [r1 , b1 ]. (Note that R1 = S is possible.) If such
a path does not exist, we must have t00 = s0 , and we let R1 = R0 . By our choice of S, we
see that either R1 = R0 , or R1 = S, or r1 ∈ MP [z2 , b1 ].
Claim 4. We may assume there is some path Ox from z1 to X 0 in B − MP [r1 , z2 ].
Proof. Since x2 ∈
/ V (MP ), we see from (3, {a1 , b1 , a2 , b2 })-connectivity of G and our choice
of S that such a path must exist if r1 ∈ MP [s, b1 ]. Suppose, then, that r1 ∈
/ MP [s, b1 ]. We
must then have R1 = R0 , and t00 = s0 . (See Figure 32.) Then for each x ∈ X 0 , we must
have either x ∈ MP [r1 , b1 ], or some 2-cut W ⊆ MP [r1 , b1 ] separates x from z1 in B. (Recall
that B is the Z-bridge of LP containing X 0 .) Hence for each x ∈ X 0 , either x is cofacial
with s0 , or W ∪ {s0 } is a 3-cut in G separating x from {b1 , a2 , b2 }. We recall that x1 is also
cofacial with s0 . We may assume, then, that s0 6= a1 ; otherwise we have (Γ, X) ∈ O3 .
Since t00 = s0 , we know P [a1 , s0 ] ⊆ V (C). Hence there must be some path S0 from
s00 ∈ P [a1 , t00 ) to s0 ∈ MP [z2 , b1 ] (by our choice of S). Choose such an S0 to minimize
P [s00 , t00 ] and MP [z2 , s0 ], so each of s00 , s0 is cofacial with t00 . For any vertex v ∈ MP [z2 , s0 ],
the set {v, b2 , s0 } is not a 3-cut separating {a1 , b1 } from {a2 } ∪ X in G, or G would have
a 4-separation (H, K) with V (H ∩ K) = {v, b2 , s0 , a2 } demonstrating that (Γ, X) satisfies
(R1). So we may assume s0 ∈
/ C[b1 , a2 ] and s0 ∈ A(P ).
Define Q := P [a1 , s00 ] ∪ S0 ∪ MP [z2 , s0 ] ∪ Qx , a path from a1 to x2 .
67
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a2
(a) Path O1
(b) Path O2
Figure 33: Forcing crossed 3-separations when t0 = x1
We claim that Q ∈ P(Γ,X) . To see this, suppose not. Then a2 is a cutvertex of LQ , and
a2 is cofacial with some vertex q ∈ V (Q). We may assume q ∈ MP [z2 , s0 ] (because of the 2cut Z in LP ). But then we have a 4-separation (M, N ) of G with V (M ∩N ) = {a2 , q, t00 , b2 },
and we find (Γ, X) satisfies property (R1). Hence we may assume a2 is not a cutvertex of
LQ , and Q ∈ P(Γ,X) .
Now (T0 ∪ Us ∪ Qb ) ⊆ LQ . If there are disjoint paths in LP from a2 to r and from b1 to
z1 respectively, each disjoint from MP [z2 , s0 ] and MP [b2 , t0 ], then we find that x1 ∈ R(Q).
Hence no such paths may exist. By our choice of Z as a maximal 2-cut, we see that there
must be some 3-cut {h1 , h2 , h3 } in LP separating {b2 , a2 , b1 , s0 } from {t0 , r, z1 , z2 }. We may
assume h1 ∈ MP [b2 , t0 ] and h2 ∈ MP [z2 , s0 ].
Now h2 is cofacial with s0 , so G has a separation (H 0 , K 0 ) of order 4 with V (H 0 ∩ K 0 ) =
{h1 , h2 , h3 , s0 }, {a1 , b1 , a2 , b2 } ⊆ V (H 0 ), and X ⊆ V (K 0 ). Since h1 , s0 ∈ C[b2 , a1 ], (Γ, X)
satisfies property (R1).
Define O1 := P [a1 , t00 ] ∪ T0 ∪ MP [t0 , t] ∪ T , a path from a1 to x1 in G. (See Figure 33(a).)
Suppose there are four disjoint paths in LP : from t0 to t, from b2 to r, from a2 to z1 ,
and from r1 to b1 in LP . These paths together with R, R1 , and Ox would show that X is
68
feasible in Γ, a contradiction. Hence no such paths may exist. There must be a separation
(M, N ) of LP of order at most 3 such that {t0 , b2 , a2 , b1 } ⊆ V (M ) and {t, r, z1 , r1 } ⊆ V (N ).
By Claim 3, there is no cutvertex of LP in MP [t0 , t]. From this, and our choice of Z as a
maximal cut, we see that |V (M ∩ N )| = 3. We may assume V (M ∩ N ) = {h1 , h2 , h3 } with
h1 ∈ MP [t0 , t], h2 ∈ MP [r1 , b1 ] ∩ MP [z2 , b1 ]. We further assume that (M, N ) is chosen to
minimize MP [z2 , h2 ].
If h2 is cofacial with t00 , there is a separation (M 0 , N 0 ) of G of order 4 such that
{a1 , b2 , a2 , b1 } ⊆ V (M 0 ) and X ⊆ V (N 0 ), and V (M 0 ∩ N 0 ) = V (M ∩ N ) ∪ {t00 }. By
maximality of Z, N 0 has disjoint paths from h1 to x1 , from t00 to h3 , and from h2 to x2 . An
application of Lemma 2.11 shows that either (Γ, X) satisfies property (R1), or (Γ, X) ∈ O2 .
So we may assume h2 is not cofacial with t00 .
From our choice of R1 , this means there is some path S0 from s00 ∈ P [a1 , t00 ) to s0 ∈
MP [r1 , h2 ). Choose such an S0 to minimize P [s00 , t00 ] and MP [r1 , s0 ]. Note by our choice of
S that s0 ∈ MP [z2 , b1 ].
Define O2 := P [a1 , s00 ] ∪ S0 ∪ MP [z2 , s0 ] ∪ Qx , a path from a1 to x2 in G.
If there are three disjoint paths, from b1 to z1 , from a2 to t, and from b2 to t0 in LP − O2 ,
then we see that T ∪R0 ⊆ LO2 , x1 ∈ R(O2 ), and X is feasible in Γ, a contradiction. Hence no
such paths exist, and there must be a 3-separation (M 0 N 0 ) of LP such that {s0 , b1 , a2 , b2 } ⊆
V (M 0 ) and {z2 , z1 , t, t0 } ⊆ V (N 0 ). We may assume V (M 0 ∩ N 0 ) = {m1 , m2 , m3 } with
m1 ∈ MP [b2 , t0 ] and m2 ∈ MP [z2 , s0 ]. (See Figure 33(b).)
Suppose m2 is a cutvertex of LP . By planarity (and since z2 ∈ A(P )), m2 is right of z2 .
But then there is a separation (L1 , L2 ) of LP with V (L1 ∩ L2 ) = {h1 , m2 , h3 }, contradicting
the choice of separation (M, N ) is to minimize MP [z2 , h2 ]. So we may assume m2 is not a
cutvertex of LP .
If m1 shares a finite face with m2 , then {m1 , m2 } is a 2-cut in LP contradicting the maximality of Z. If {m1 , m3 } separates {b2 , a2 , b1 } from {t0 , t, r0 } in LP , then m3 ∈ MP [z1 , b1 ].
By maximality of Z, we find m3 ∈ MP [z1 , z2 ), and we may assume m2 = z2 , sharing a finite
face with m3 . So we may in any case assume m3 shares a finite face of LP with each of m1
and m2 .
69
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(c)
Figure 34: Cases of crossed 3-separations
Note that the vertices (b2 , m1 , t0 , h1 , t, z2 , m2 , s0 , h2 , b1 ) occur in order along MP . Observe from the planarity of LP that one of the following must hold:
(a) {h1 , m2 } is a 2-cut in LP (Figure 34(a)), or
(b) {m1 , h1 } is a 2-cut in LP , {h1 , m2 , m3 } is a 3-cut in LP , and h3 is cofacial with each
of m1 , h2 (Figure 34(b)), or
(c) {m2 , h2 } is a 2-cut in LP , {h1 , m2 , h3 } is a 3-cut in LP , and m3 is cofacial with each
of m1 , h2 (Figure 34(c)).
But (a) contradicts the choice of Z. Hence (b) or (c) must hold, and we have some
v ∈ {h3 , m3 } which is cofacial with each of m1 and h2 .
Suppose {m1 , v, h2 } = {b2 , a2 , b1 }. By (3, {a1 , b1 , a2 , b2 })-connectivity of G, {b2 , a2 , b1 }
is not a 3-cut in LP , so h3 , m3 ∈ C[b1 , b2 ]. In case (b), h1 is a cutvertex of LP , contradicting
Claim 3. In case (c), m2 is a cutvertex of LP , contradicting our choice of the separation
(M, N ) (minimizing MP [z2 , h2 ]).
We may now assume that {m1 , v, h2 } is a 3-cut in LP . Note that h2 is cofacial with
some vertex of P [a1 , s0o ] = C[s00 , a1 ]. So we have a separation (E, F ) of G with V (E ∩ F ) =
{m1 , h2 , v1 , s00 }, {a1 , b1 , a2 , b2 } ⊆ V (E), and X 0 ⊆ V (F ). Since m1 , s00 ∈ C[b2 , a1 ], we find
70
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a2
(a)
(b)
Figure 35: Where a2 is a cutvertex of LQ
that (Γ, X) satisfies property (R1).
3.5.3
The case that T falls to the left of P
Define paths Q1 and Q:
For any path O from si ∈ {s1 , . . . sk } to X 0 in B − {z1 , z2 },
let m(O) ∈ V (MP ) be the vertex of O ∩ MP which lies leftmost along MP . Choose a
path Q1 from si ∈ {s1 , . . . sk } to X 0 in B − {z1 , z2 } minimizing MP [m(Q1 ), z2 ]. Define
Q := P [a1 , s0 ] ∪ Si ∪ Q1 , a path from a1 to X 0 . Recalling that path R is from r 0 ∈ P (s0 , x1 ]
to r ∈ MP (t, z1 ), we note that (T ∪ R) ⊆ LQ .
Claim 5. We may assume Q ∈ P(Γ,X) .
Proof. We first show that Q1 is disjoint from C[b1 , b2 ].
If not, we have MP (z1 , z2 ) ∩
C[b1 , b2 ] 6= ∅. This forces {z1 , z2 } ⊆ V (C). So one of z1 or z2 is a cutvertex of LP
separating X 0 from a2 , violating our assumption that X 0 ⊆ A(P ).
Now suppose a2 is a cutvertex of LQ , and hence a2 is cofacial with some vertex of Q.
Noting that Q ∩ LP ⊆ B, we find that a2 , z1 , and z2 are cofacial. (See Figure 35.) Now
z2 6= b1 . To see this, suppose z2 = b1 . Since G is (3, {a1 , b1 , a2 , b2 })-connected, {b1 , a2 } is
not a 2-cut in G, and we must have z1 ∈ C(b1 , a2 ). So z1 is a cutvertex of LP separating
X 0 from a2 . This contradicts our observation that x2 ∈ A(P ).
Suppose some vertex p ∈ P [a1 , x1 ) is cofacial with both z2 and b2 . Then {b2 , p, z2 , a2 }
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Figure 36: Defining Z 0 (U )
is a 4-cut in G, and we find (Γ, X) satisfies property (R1). Hence we may assume there is
no such vertex p, and we must have paths T0 from t00 ∈ P [a1 , s0 ) to t0 ∈ MP (b2 , t] and S0
from s00 ∈ P (t00 , s] to s0 ∈ MP (z2 , b1 ]. Choose such paths to minimize P [t00 , s0 ], P [t00 , s00 ],
MP [t0 , t], and MP [s0 , b1 ].
Let O := P [a1 , t00 ] ∪ T0 ∪ MP [t0 , t] ∪ T ∪ P [t, x1 ]. (Here we are using a routing similar
to that in Proposition 3.10.) If there are disjoint paths from b2 to z1 , from a2 to z2 ,
and from b1 to s0 in LP − V (MP [t0 , t]), then X is feasible in Γ, a contradiction. Hence
such paths do not exist, and there must be some separation (M, N ) of LP of order at
most 3, with {t, b2 , a2 , b1 } ⊆ V (M ) and {t0 , z1 , z2 , s0 } ⊆ V (N ). By maximality of Z,
we have |V (M ∩ N )| = 3. Moreover, since G is (3, {a1 , b1 , a2 , b2 })-connected, we have
V (M ∩ N ) = {m1 , m2 , a2 } with m1 ∈ MP [t0 , t] and m2 ∈ MP [s0 , b1 ]. Hence {m1 , a2 , z2 } is a
3-cut in LP . By our choice of T and T0 , we have m1 cofacial with s0 , so there is a separation
(G1 , G2 ) of G with V (G1 ∩ G2 ) = {m1 , s0 , z2 , a2 }, {a1 , b1 , a2 , b2 } ⊆ V (G1 ), X 0 ⊆ V (G2 ).
Hence (Γ, X) satisfies (R1).
Define general Z 0 (U ):
Given any path U from u0 ∈ P (s0 , x1 ] to u ∈ MP [z1 , s) such that
U * LQ , we define a 2-cut Z 0 (U ) of LP as follows. (See Figure 36.)
Since T ∪ R ⊆ LQ and z1 ∈ LQ , but U * LQ , we must have MP (z1 , u] ∩ Q 6= ∅.
72
Recalling our choice of Q1 ⊆ Q to minimize MP [m(Q1 ), z2 ], we have m(Q1 ) ∈ MP (z1 , u].
Moreover, there is no path from {s1 , . . . sk } to x2 in B − {z1 , z2 } disjoint from MP (z1 , u].
Since x2 ∈
/ V (MP ), we find that there must be some 2-cut Z 0 (U ) ⊆ (V (MP [z1 , u]) ∪ {z2 })
separating {s1 , . . . sk } from x2 in B. Again from our choice of Q1 , we may assume that
Z 0 (U ) separates X 0 from s1 in B. Let Z 0 (U ) = {z10 , z20 } with z10 left of z20 . We may have
Z 0 (U ) ⊆ V (MP [z1 , u]), as illustrated in Figure 36(a), or z20 = z2 , as in Figure 36(b).
Claim 6. We may assume x1 ∈ LQ .
Proof. Suppose x1 ∈
/ LQ . Since (T ∪ R) ⊆ LQ , it must be that R falls to the left of P . (See
Figure 37.) From (3, {a1 , b1 , a2 , b2 })-connectivity of G, together with Lemma 3.3 (limiting
paths within bridges) we know there are at least two internally disjoint paths from x1 to
MP . Let U1 and U2 be paths from x1 to u1 , u2 ∈ MP respectively, chosen to minimize
MP [z1 , u1 ] and MP [u1 , u2 ]. Since x1 ∈
/ LQ , we have U1 * LQ . Let Z 0 := Z 0 (U1 ), and label
Z 0 = {z10 , z20 }, with z10 left of z20 .
We now proceed to show that there is some path R0 from r00 ∈ P [r 0 , x1 ) to r0 ∈
MP (z1 , u1 ]. Suppose first that Z 0 ⊆ V (MP [z1 , u1 ]) (Figure 37(a)). Then from the (3, {a1 , b1 , a2 , b2 })connectivity of G, Z 0 is not a 2-cut in G. Hence there must be a path R0 as desired. Now
suppose z20 = z2 (Figure 37(b)). If no path R0 exists, then Z 0 ∪ {z1 } is a 3-cut in G,
separating X 0 from s1 . Furthermore, z1 is cofacial with x1 . Hence we have a 4-separation
(M, N ) of G with V (M ∩ N ) = {z1 , x1 , z10 , z20 }, X ⊆ V (N ), and {a1 , b1 , a2 , b2 } ⊆ V (M ). An
application of Lemma 2.11 gives (Γ, X) ∈ O2 , or (Γ, X) is reducible. Hence we may assume
the path R0 exists. Choose such a path R0 to minimize P [r 0 , r00 ] and MP [z1 , r00 ], so z1 and
r00 are cofacial.
Now choose a path S0 from s00 ∈ P [a1 , r 0 ) to s0 ∈ MP [u2 , s1 ] to minimize MP [u2 , s0 ]
and, subject to this, to minimize P [s00 , r 0 ] (where S0 = S1 is possible). Choose a path U
from u0 ∈ P (r00 , x1 ] to u ∈ MP [u, s1 ] to minimize P [r00 , u0 ] and MP [u, s0 ] (where U = U2 is
possible). Note that each vertex of MP [u, s0 ] is cofacial with r 0 .
Define O := P [a1 , s00 ] ∪ S0 ∪ MP [u, s0 ] ∪ U ∪ P [u0 , x1 ] (Figure 37(c)). If there is a path
QO from r0 to z2 in B − MP [u, s0 ], then we find R ∪ R0 ∪ Q0 ⊆ LO . Hence z1 ∈ R(O).
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Figure 37: Where x1 not in LQ
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But R(P ) ⊆ R(O), so this violates the choice of P as a maximizing path. So no such path
QO may exist, and there must be some cutvertex v of B with v ∈ MP [u, s0 ]. Now there is
a 3-separation (M, N ) in G with V (M ∩ N ) = {z1 , v, r 0 }, X ⊆ V (N ), and {a1 , b1 , a2 , b2 } ⊆
V (M ), and we have (Γ, X) ∈ O2 .
Claim 7. We may assume x1 ∈
/ MQ .
Proof. Suppose x1 ∈ MQ . Here we nearly repeat the arguments of Claim 6, with a slightly
different choice of paths.
Consider first the case that x1 is cofacial with z1 . If x1 is cofacial with one of s0
or s, we have a 4-separaton (M, N ) in G with V (M ∩ N ) = {z1 , z2 , x1 , s} or V (M ∩
N ) = {z1 , z2 , x1 , s0 }, X ⊆ V (N ), and {a1 , b1 , a2 , b2 } ⊆ V (M ). Applying Lemma 2.11 (4separations in G), we find that (Γ, X) ∈ O2 , or (Γ, X) is reducible. Hence we may assume
x1 is not cofacial with either s0 or s, and there is some path U1 from u01 ∈ P (s0 , x1 ) to
u1 ∈ MP [z1 , s), falling to the right of P . Choose such a path to minimize P [u01 , x1 ] and
MP [z1 , u1 ].
Since x1 ∈ MQ , we have U1 * LQ . Let Z 0 := Z 0 (U1 ), and label Z 0 = {z10 , z20 }, with z10
left of z20 . If Z 0 ⊆ V (MP [z1 , u1 ]) (Figure 38(a)), we see from our choice of U1 that Z 0 ∪ {x1 }
is a 3-cut in G demonstrating that (Γ, X) ∈ O2 . Therefore we may assume z20 = z2 (Figure
38(b)). But then Z 0 ∪{z1 } is a 3-cut in G, separating X 0 from s1 , and we have a 4-separation
(M, N ) of G with V (M ∩ N ) = {z1 , x1 , z10 , z20 }, X ⊆ V (N ), and {a1 , b1 , a2 , b2 } ⊆ V (M ). An
application of Lemma 2.11 gives (Γ, X) ∈ O2 , or (Γ, X) is reducible.
Now consider the case where x1 is not cofacial with z1 (Figure 38(c)). Because x1 ∈
V (MQ ), let R0 be a path from r00 ∈ P [r 0 , x1 ) to r0 ∈ MP (z1 , s], chosen to minimize P [r 0 , r00 ]
and MP [z1 , r00 ], so z1 and r00 are cofacial. From (3, {a1 , b1 , a2 , b2 })-connectivity of G, together
with Lemma 3.3 (limiting paths within bridges) we know there are at least two internally
disjoint paths from x1 to MP . From this together with planarity, we may assume r0 ∈
/
{s1 , . . . , sk }.
Now choose a path S0 from s00 ∈ P [a1 , r 0 ) to s0 ∈ MP (r0 , s1 ] to minimize MP [r0 , s0 ] and,
subject to this, to minimize P [s00 , r 0 ] (where S0 = S1 is possible). Choose a path U from
75
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Figure 38: Where x1 ∈ MQ
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(a) External
(b) Internal
Figure 39: Defining separation (Hz , Mz )
u0 ∈ P (r00 , x1 ] to u ∈ MP (r0 , s1 ] to minimize MP [u, s0 ] and then P [r00 , u0 ]. Note that each
vertex of MP [u, s0 ] is cofacial with r00 .
Define O := P [a1 , s00 ] ∪ S0 ∪ MP [u, s0 ] ∪ U ∪ P [u0 , x1 ]. If there is a path QO from r0 to
z2 in B − MP [u, s0 ], then we find R ∪ R0 ∪ Q0 ⊆ LO . Hence z1 ∈ R(O). But R(P ) ⊆ R(O),
so this violates the choice of P as a maximizing path. So no such path QO may exist, and
there must be some cutvertex z3 of B with v ∈ MP [u, s0 ]. Now there is a 3-separation
(M, N ) in G with V (M ∩ N ) = {z1 , z3 , r00 }, X ⊆ V (N ), and {a1 , b1 , a2 , b2 } ⊆ V (M ), and
we have (Γ, X) ∈ O2 .
Refine choice of path R: Assume R is chosen such that R ⊆ MQ . By Claims 6 and 7,
we may assume R is from r 0 ∈ P (s0 , x1 ) to r ∈ MP (t, s0 ), and falls to the right of P .
Define Paths Ut and Ur :
From (3, {a1 , b1 , a2 , b2 })-connectivity of G, {r, r 0 } is not a 2-cut
in G, so there must be some path Ut0 from u0t ∈ P (r 0 , x1 ] to ut ∈ MP [t, r). Choose such a
path Ut0 to minimize MP [t, ut ] and then P [r 0 , u0t ], and let Ut = Ut0 ∪P [u0t , x1 ]. Similarly, {t, t0 }
is not a 2-cut in G, so there must be some path Ur0 from u0r ∈ P (t0 , x1 ] to ur ∈ MP (t, r].
Choose such a path Ur0 to minimize MP [ur , r] and then P [t0 , u0r ], and let Ur = Ur0 ∪ P [u0r , x1 ].
77
Define separation (Hz , Mz ):
We proceed to define a 3-separation (Hz , Mz ) of G with
X 0 ⊆ V (Mz ), {a1 , b1 , a2 , b2 , x1 } ⊆ V (Hz ), and V (Hz ∩ Mz ) = {uz , vz , z2 }.
If z1 is cofacial with some vertex uz ∈ {s, s0 }, then we have a 3-cut Z ∪ {uz } in G,
separating X 0 from {a1 , b1 , a2 , b2 , x1 }. Let (Hz , Mz ) be the associated 3-separation, with
vz = z1 . We will call this as an external separation (Figure 39(a)).
Now suppose z1 is not cofacial with either of s0 , s. Then path Us (defined in Section 3.5.1)
has u0s ∈ P (s0 , x1 ] and us ∈ MP (z1 , s), as in Figure 39(b). From our choice of R ⊆ MQ , we
know Us * LQ . Let Z 0 := Z 0 (Us ), and label Z 0 = {z10 , z20 }, with z10 left of z20 . Since G is
(3, {a1 , b1 , a2 , b2 })-connected, and by our choice of Us to minimize MP [z1 , us ], we see that
Z 0 * MP [z1 , us ], so we must have z20 = z2 . Again by our choice of Us , we have MP [z1 , us ]
contained in a facial cycle of G, so there is a 3-cut {z1 , z10 , z2 } in G separating X 0 from
{a1 , b1 , a2 , b2 , x1 }. Let (Hz , Mz ) be the associated 3-separation, with vz = z1 , uz = z10 . We
call this as an internal separation.
Claim 8. If z2 6= b1 , we may assume z1 is cofacial with one of s0 or s.
Proof. Suppose z2 6= b1 , but z1 is cofacial with neither s nor s0 . Then u0s ∈ P (s0 , x1 ]
and us ∈ MP (z1 , s), and (Hz , Mz ) is an internal separation. Since G is (3, {a1 , b1 , a2 , b2 })connected, and by planarity of G, we note that there is no cutvertex of B in MP [z1 , us ]. So
we have two disjoint paths in B: Qb = MP [z1 , us ] and Q0x from z2 to X 0 .
Let O1 := P [a1 , s0 ] ∪ S ∪ MP [s, z2 ] ∪ Qx , a path from a1 to x2 . Suppose there are disjoint
paths in LP from b1 to z1 and from a2 to ur , each disjoint from MP [b2 , t] and z2 . These
paths, together with MP [b1 , t] ∪ T ∪ Us ∪ MP [z1 , us ] and Ur , show that X is feasible in Γ,
as shown in Figure 40(a).
Hence no such paths exist. There must be a separation (M, N ) of LP of order at most
three such that {b1 , a2 , b2 } ⊆ V (M ), {z1 , ur , t} ⊆ V (N ), and z2 ∈ V (M ∩ N ). From our
choice of Z as a maximal cut, we see that |V (M ∩ N )| = 3. We may assume V (M ∩ N ) =
{h1 , h2 , z2 } with h1 ∈ MP [b2 , t] (Figure 40(b)).
If this extends to a 4-separation (M 0 , N 0 ) of G with V (M 0 ∩ N 0 ) = {h1 , h2 , z2 , v} for
some v ∈ P [a1 , s0 ], then by Lemma 2.11, (Γ, X) satisfies property (R1) or (Γ, X) ∈ O2 . So
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(b) Separation (M, N )
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Figure 40: Example of z2 6= b1 , not cofacial s or s0
79
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we may assume there is no such 4-separation. By our choice of S, there are paths T0 from
t00 ∈ P [a1 , s0 ) to t0 ∈ MP (h1 , t] and S0 from s00 ∈ P (t00 , s0 ] to s0 ∈ MP (z2 , b1 ]. Choose such
paths T0 , S0 to minimize P [t00 , s0 ], MP [t0 , t], P [s00 , s0 ], and MP [z2 , s0 ]. Note from this choice
that each vertex of MP [t0 , t] is cofacial with some vertex of P [s00 , s0 ]. Furthermore, each
vertex of P [s00 , s0 ] is cofacial with z2 .
Let O2 := P [a1 , t00 ] ∪ T0 ∪ MP [t0 , t] ∪ T ∪ P [t0 , x1 ], from a1 to x1 in G. Suppose there are
disjoint paths from b1 to z1 and from a2 to z2 , each disjoint from MP [t0 , t] and MP [s0 , b1 ].
These paths, together with S ∪ S0 , Qb , and Q0x show that X is feasible in Γ, a contradiction.
Hence no such paths exist, and there must be a separation (M 0 N 0 ) of LP of order at most
three such that {t0 , b1 , a2 , b2 } ⊆ V (M 0 ) and {t, z1 , z2 , s0 } ⊆ V (N 0 ). From our choice of Z
as a maximal cut, |V (M 0 ∩ N 0 )| = 3. We may assume V (M 0 ∩ N 0 ) = {m1 , m2 , m3 } with
m1 ∈ MP [t0 , t] and m2 ∈ MP [s0 , b1 ] (Figure 40(c)).
We now have 3-cuts {h1 , h2 , z2 } and {m1 , m2 , m3 } in LP , with the vertices
b2 , h1 , t0 , m1 , t, z2 , s0 , m2 , b1 in order along MP . Observe from the planarity of LP that
one of the following must hold:
(i) {m1 , z2 } is a 2-cut in LP
(ii) There is some v ∈ {h2 , m3 } such that {m1 , z2 , v} is 3-cuts in LP .
But (i) contradicts the choice of Z, so (ii) must hold. Note that m1 is cofacial with some
vertex p ∈ P [s00 , s0 ]. So we have a separation (E, F ) of G with V (E ∩ F ) = {m1 , z2 , v, p},
{a1 , b1 , a2 , b2 } ⊆ V (E), and X 0 ⊆ V (F ). Applying Lemma 2.11 we find that (Γ, X) ∈ O2 ,
or (Γ, X) is reducible.
Define 2-cut W : Since we assume x1 ∈
/ R(Q), there is some 2-separation (L01 , L02 ) of
LQ with x1 ∈ V (L01 ) and {b1 , a2 , b2 } ⊆ V (L02 ). Let W = V (L01 ∩ L02 ) = {w1 , w2 }. Since
T ∪ R ⊆ LQ , we may assume w1 ∈ MQ [b2 , t] = MP [b2 , t] and w2 ∈ MQ [r, b1 ]. Assume also
that W is chosen to minimize MQ [w1 , t] and MQ [w2 , b1 ].
Observe that if w2 ∈ MQ [z2 , b1 ] = MP [z2 , b1 ] then W is a 2-cut in LP separating a2
from x2 , contradicting our choice of Z as a maximal 2-cut. Hence w2 ∈ MQ [r, z2 ). Note
also that w1 = b2 is possible. In this case, w2 is a cutvertex of LQ .
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Figure 41: 2-cut in LQ corresponding to 3-cut in LP
Define vertices z0 and z3 :
Let z0 ∈ V (MP [b2 , w1 ] ∩ C[b1 , b2 ]) be chosen to minimize
MP [z0 , w1 ]. Note that either z0 is a cutvertex of LP , or z0 = b2 . Let z3 ∈ V (MP [z2 , b1 ] ∩
C[b1 , b2 ]) be chosen to minimize MP [z2 , z3 ]. Note that either z3 is a cutvertex of LP , or
z3 = b1 .
Claim 9. We may assume w2 ∈ MQ [r, z1 ).
Proof. If Claim 9 is false, there is a 3-separation (L1 , L2 ) of LP with L1 ∩ L2 = {w1 , w2 , z2 },
{b1 , a2 , b2 } ⊆ V (L1 ), {t, r} ∪ X 0 ⊆ V (L2 ). (See Figure 41(a).) Suppose this corresponds
to a 4-separation (M, N ) of G with V (M ∩ N ) = {w1 , w2 , z2 , v} for some v ∈ P [a1 , s0 ],
with {a1 , b1 , a2 , b2 } ⊆ V (M ) and X ⊆ V (N ). An application of Lemma 2.11 (4-separations
in G) gives (Γ, X) ∈ O2 , or (Γ, X) is reducible. Hence we may assume there is no such
4-separation of G.
By our choice of S, there are paths T0 from t00 ∈ P [a1 , s0 ) to t0 ∈ MP (w1 , t] and S0
from s00 ∈ P (t00 , s0 ] to s0 ∈ MP (z2 , b1 ] (Figure 41(b)). Choose T0 to minimize P [a1 , t00 ] and
MP [b2 , t0 ]. Subject to this, choose S0 to minimize P [t00 , s00 ] and MP [s0 , b1 ].
Let O := P [a1 , t00 ] ∪ T0 ∪ MP [t0 , t] ∪ T ∪ P [t0 , x1 ], a path from a1 to x1 . From our choice
of w1 , and assumption that Claim 9 is false, we see that there is a path from w1 to z1 in L2
disjoint from MP [t0 , t], and hence disjoint from O. Moreover, this path is disjoint from a2 ;
81
otherwise we have a2 = w2 = z1 , and a2 is a cutvertex of LP . So there is a path from b2 to
z1 , disjoint from O and C[b1 , a2 ], and we find that S1 ∪ S0 ⊆ LO , and x2 ∈ (LO \ MO ).
Recall that we have disjoint paths the paths Qb = MP [z1 , us ] and Qx from z2 to x2
in B. If L1 contains a path from a2 to z2 which is disjoint from MP [b2 , w1 ] ∪ MP [s0 , b1 ],
then we find that X is feasible in Γ. Hence no such path exists, and there must be a 2-cut
Z 0 = {z10 , z20 } in LP separating x2 from a2 , with z10 ∈ MP [b2 , w1 ] and z20 ∈ MP [s0 , b1 ]. But
such a cut also separates X 0 from a2 , contradicting our choice of Z.
Define C, Cw :
We call a pair of paths (Ti , Ri ) consistent if the following conditions hold:
1. Ti is from t0i ∈ P [a1 , ri0 ) to ti ∈ MP (b2 , t], and
2. MP [ti , ut ] ∩ V (C) = ∅, and
3. Ri is from ri0 ∈ P (t0i , r 0 ] to ri ∈ MP [z1 , b1 ].
Let C be the collection of all consistent pairs of paths. Let Cw ⊆ C be the collection of all
consistent pairs (Ti , Ri ) such that ti ∈ MP (w1 , t].
Claim 10. If w2 ∈ V (C), then for all (Ti , Ri ) ∈ C, we have ri = z1 .
Proof. Suppose there is some pair of paths (T0 , R0 ) ∈ C with r0 ∈ MP (z1 , b1 ]. Letting
O := P [a1 , t00 ] ∪ MP [t0 , ut ] ∪ Ut , we have R ∪ R0 ⊆ LO . From our choice of Z, we find that
no 2-cut of LP contained in MP [s, b1 ] ∪ {w2 } separates X 0 from a2 . Hence there is a path
from a2 to z1 in LP which is disjoint from O. So z1 ∈ R(O). Since R(P ) ⊆ R(O) and
z1 ∈
/ R(P ), this contradicts the assumption that P is a maximal path.
Define paths S0 and T0 :
If there is a path S0 from s00 ∈ P [a1 , s0 ) to s0 ∈ MP [z2 , b1 ],
choose such an S0 to minimize P [s00 , s0 ] and MP [s, s0 ], so each of s00 , s0 is cofacial with s0 .
If there is a path T0 from t00 ∈ P [a1 , s0 ) to t0 ∈ MP [b2 , t], choose such a T0 to minimize
P [t00 , s0 ] and MP [t0 , t], so each of t00 , t0 is cofacial with s0 . Note by our choice of S and
T that if s0 ∈
/ V (C), both paths S0 and T0 exist. If s0 = a1 , neither path exists, and if
s0 ∈ V (C) \ {a1 }, exactly one of S0 , T0 exists.
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Claim 11. We may assume {w2 , s0 } * V (C).
Proof. Let {w2 , s0 } ⊆ V (C). Then w2 ∈ C[b1 , a2 ], is shown in Figure 42.
Since s 6= z1 , Claim 10 tells us that there is no path T0 for which (T0 , S) ∈ C. From
our choice of T and z0 , we find that either z0 shares a finite face with s0 , or z0 = b2 with
a1 ∈
/ C(b2 , s0 ). Since z0 , s0 ∈ V (C), there is a 2-separation (Gl , Gr ) of G with V (Gl ∩ Gr ) =
{z0 , s0 }, b2 ∈ V (Gl ), and {b1 , a2 } ∪ X ⊆ V (Gr ).
Suppose a1 ∈ V (Gl ). Then we have a 4-separation (M, N ) of G with V (M ∩ N ) =
{z0 , s0 , b1 , a2 }, such that {a1 , b1 , a2 , b2 } ⊆ V (M ) and X ⊆ V (N ). If either z0 6= b2 or
s0 6= a1 , then the separation is non-trivial and (Γ, X) satisfies property (R1). So we may
assume that one of the following holds:
(a1) a1 ∈
/ V (Gl ), so a1 ∈ C(s0 , b1 ), or
(a2) z0 = b2 and s0 = a1 .
If s0 shares a finite face with z3 , we have a 4-separation (M 0 , N 0 ) of G with V (M 0 ∩N 0 ) =
{s0 , z3 , a2 , b2 }, such that {a1 , b1 , a2 , b2 } ⊆ V (M 0 ) and X ⊆ V (N 0 ). Again (Γ, X) satisfies
property (R1). Hence we may assume s0 does not share a finite face with z3 . So one of the
following holds:
(b1) the path S0 exists, with s00 ∈ P [a1 , s0 ) to s0 ∈ MP [z2 , z3 ), or
(b2) s0 = a1 and z3 = b1 .
Case 1
(b1) is satisfied (Figure 42(a)).
Let O := P [a1 , s00 ] ∪ S0 ∪ MP [z2 , s0 ] ∪ Qx , a path from a1 to x2 . By our choice of z3 , we
find O ∩ C[b1 , b2 ] = ∅. Note that MP [b2 , t] ∪ T ∪ S ⊆ LO .
Suppose there is a path Ob from b1 to z1 in LP which is disjoint from MP [z2 , s0 ] ∪
C[a2 , w2 ]. Then we have Ob ⊆ LO . Furthermore, the path Ur ∪ MP [ur , w2 ] ∪ C[a2 , w2 ]
shows that x1 ∈ R(O). This contradicts the assumption that X is infeasible in Γ, and
hence cannot occur.
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z1
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a2
(b)
Figure 42: Where {w2 , s0 } ⊆ V (C).
84
So there must be some 2-cut V = {v1 , v2 } in LP with v1 ∈ C[a2 , w2 ] and v2 ∈ MP [z2 , s0 ],
separating b1 from z1 . From our choice of S0 , we have v2 cofacial with s0 , and there is a
4-separation (M, N ) of G with V (M ∩N ) = {v1 , v2 , s0 , z0 }, such that {a1 , b1 , a2 , b2 } ⊆ V (M )
and X ⊆ V (N ). Hence (Γ, X) satisfies property (R1).
Case 2
(b2) is satisfied: s0 = a1 and z3 = b1 .
Since (a1) cannot hold when s0 = a1 , we must also have z0 = b2 (Figure 42(b)).
Suppose the path Us has u0s = s0 . Then we see that {z1 , z2 , a1 } is a 3-cut in G separating
{b1 , a2 , b2 } from X 0 , and {w1 , w2 , a1 } is a 3-cut in G separating {b1 , a2 , b2 } from x1 . That
is, (Γ, X) ∈ O3 . Hence we may assume u0s 6= s0 .
Now suppose there is a path Sc ⊆ C from s0 to sc ∈ MP (us , b1 ). Define O1 := Sc ∪
MP [s, z2 ] ∪ Qx , a path from a1 to x2 . Note that MP [b2 , t] ∪ T ∪ Us ⊆ LO . Applying the same
argument as we did for path O above, we find that there must be some 2-cut V = {v1 , v2 }
in LP with v1 ∈ C[a2 , w2 ] and v2 ∈ MP [z2 , sc ], separating b1 from z1 . Again v2 is cofacial
with s0 , leading to a 4-separation (M, N ) of G which shows that (Γ, X) satisfies property
(R1).
Now we may assume there is no path Sc . Applying Proposition 2.9, we assume there
is no edge a1 b1 , so we must have s ∈ V (C) and us = s. From Claim 10, we find that
(T, Us ) ∈
/ C, so we must have t0 ∈ P [us , x1 ).
We now have the following in G:
• A finite facial cycle D with w2 , s0 ∈ V (D)
• A separation (H1 , M1 ) of G with V (H1 ∩ M1 ) = {s, z1 , z2 }, s ∈ V (C ∩ D), z1 ∈ V (D),
z2 ∈ V (C), {a1 , a2 , b1 , b2 } ⊆ V (H1 ),
• A separation (H2 , M2 ) of G with V (H2 ∩M2 ) = {w2 , t0 , b2 }, w2 ∈ V (C ∩D), t0 ∈ V (D),
b2 ∈ V (C), {a1 , a2 , b1 , b2 } ⊆ V (H2 )
where X ⊆ V (M1 ∪ M2 ∪ D ∪ C[z2 , b1 ] ∪ C[b2 , b2 ]), X ∩ M1 6= ∅, and X ∩ M2 6= ∅. So we
have (Γ, X) ∈ O6 .
Claim 12. We may assume w2 ∈
/ V (C).
85
Proof. Suppose w2 ∈ V (C). From Claim 11, we may assume s0 ∈
/ V (C). So in particular,
paths S0 and T0 both exist.
Consider first the case that s0 ∈ MP [z2 , z3 ). Let O := P [a1 , s00 ] ∪ S0 ∪ MP [z2 , s0 ] ∪ Qx ,
a path from a1 to x2 . Note that MP [b2 , t] ∪ T ∪ S ⊆ LO .
Suppose there is a path Ob from b1 to z1 in LP which is disjoint from MP [z2 , s0 ] ∪
C[a2 , w2 ]. Then we have Ob ⊆ LO . Furthermore, the path Ur ∪ MP [ur , w2 ] ∪ C[a2 , w2 ]
shows that x1 ∈ R(O). This contradicts the assumption that X is infeasible in Γ, and
hence cannot occur.
So there must be some 2-cut V = {v1 , v2 } in LP with v1 ∈ C[a2 , w2 ] and v2 ∈ MP [z2 , s0 ],
separating z3 from z1 . From our choice of S0 , we have v2 cofacial with s0 , and there is a
4-separation (M, N ) of G with V (M ∩N ) = {v1 , v2 , s0 , z0 }, such that {a1 , b1 , a2 , b2 } ⊆ V (M )
and X ⊆ V (N ). Hence (Γ, X) satisfies property (R1).
Now consider the case that s0 ∈ MP [z3 , b1 ]. By our choice of S0 , and assumption
that there is no edge a1 b1 , we note that either z3 6= b1 or s00 6= a1 . We may assume
t0 6= b2 ; otherwise, we have a 3-cut {b1 , b2 , s0 } in G, and a 4-separation (M, N ) of G with
V (M ∩ N ) = {b1 , a2 , b2 , s0 } demonstrating that (Γ, X) satisfies property (R1).
From Claim 10, and since s 6= z1 , we see that (T0 , S) ∈
/ C. Hence there must be some
vertex v3 ∈ MP [t0 , ut ] ∩ C[b1 , b2 ], separating t0 from ut in LP . By planarity, this forces
w1 ∈ C[b1 , b2 ]. By our choice of T0 and T , w1 is cofacial with s0 . Hence {w1 , s0 , z3 } is a
3-cut in G, and there is a 4-separation (M, N ) of G with V (M ∩ N ) = {w1 , s0 , z3 , a2 }, such
that {a1 , b1 , a2 , b2 } ⊆ V (M ) and X ⊆ V (N ). Hence (Γ, X) satisfies property (R1).
Claim 13. If S0 exists with s0 ∈ MP [z2 , z3 ), then there is a 3-cut {h1 , h2 , h3 } in LP
separating {b2 , a2 , b1 , s0 } from {w1 , w2 , z1 , z2 }, with h1 ∈ MP [b2 , w1 ] and h2 ∈ MP [z2 , s0 ].
Moreover, T0 exists with t0 ∈ MP (h1 , w1 ].
Proof. Let Os := P [a1 , s00 ] ∪ S0 ∪ MP [z2 , s0 ] ∪ Qx , from a1 to x2 . Suppose there are disjoint
path from a2 to w2 and from b1 to z1 in LP , each disjoint from MP [b2 , w1 ] and MP [z2 , s0 ].
Together with T ∪ S and Ur , these would show x1 ∈ R(Os ), contradicting the assumption
that X is infeasible in Γ. Hence no such paths may exist.
86
By our choice of Z as a maximal 2-cut, we see that there must be some 3-cut {h1 , h2 , h3 }
in LP separating {b2 , a2 , b1 , s0 } from {w1 , w2 , z1 , z2 }. We may assume h1 ∈ MP [b2 , w1 ] and
h2 ∈ MP [z2 , s0 ]. From our choice of S0 , we see that h2 is cofacial with s0 . If h1 is also
cofacial with s0 , we have a 4-cut {h1 , h2 , h3 , s0 }. Applying Lemma 2.11 to the associated
4-separation, we find that (Γ, X) ∈ O2 or (Γ, X) satisfies property (R1).
Hence we may assume h1 is not cofacial with s0 . From our choice of T , we see that there
must be a path T0 with t0 ∈ MP (h1 , w1 ].
Claim 14. If T0 exists with t0 ∈ MP (z0 , t], then there is a 3-cut {m1 , m2 , m3 } in LP
separating {t0 , b2 , a2 , b1 } from {ut , w2 , z1 , z2 }, with m1 ∈ MP [t0 , ut ] and m2 ∈ MP [z2 , b1 ].
Moreover, S0 exists with s00 ∈ P [a1 , t00 ), s0 ∈ MP [z2 , m2 ).
Proof. Let Ot := P [a1 , t00 ] ∪ T0 ∪ MP [t0 , ut ] ∪ Ut , from a1 to x1 . Suppose there are disjoint
path from b2 to w2 and from a2 to z1 in LP , each disjoint from MP [t0 , ut ] ∪ MP [z2 , b1 ].
Together with R ∪ S, these would show x2 ∈ R(Ot ), contradicting the assumption that X
is infeasible in Γ. Hence no such paths may exist.
By our choice of Z as a maximal 2-cut, we see that there must be some 3-cut {m1 , m2 , m3 }
in LP separating {t0 , b2 , a2 , b1 } from {ut , w2 , z1 , z2 }. We may assume m1 ∈ MP [t0 , ut ] and
m2 ∈ MP [z2 , b1 ], and choose such a cut to minimize MP [b2 , m1 ], and subject to this to
minimize MP [z2 , m2 ].
Consider the case that m1 is cofacial with s0 . If m2 is also cofacial with s0 , we have a
4-cut {m1 , m2 , m3 , s0 }. Applying Lemma 2.11 to the associated 4-separation, we find that
(Γ, X) ∈ O2 or (Γ, X) satisfies property (R1). So we may assume m2 is not cofacial with
s0 , and by our choice of S, S0 exists with s0 ∈ MP [z2 , m2 ).
Now consider the case that m1 is not cofacial with s0 . By our choice of T and T0 , we must
have m1 ∈ MP (t, ut ]. Letting Bw be the W -bridge of G containing MP (w1 , w2 ), we see that
m1 is a cutvertex of Bw . Since m1 is not cofacial with s0 , we may assume t ∈ MP [w1 , m1 ).
Now w1 is cofacial with some vertex of P [a1 , t00 ]. Let t0v be such a vertex, chosen to
minimize P [t0v , x1 ]. Then there is a path Tv from t0v to tv ∈ MP (w1 , m1 ). Let Tv be chosen
to minimize P [w1 , tv ].
87
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Figure 43: Where S0 and T0 both exist
If m2 is cofacial with t0v , we have a 4-cut {w1 , m2 , m3 , t0v } in G. (By Claim 12, w2 ∈
/ V (C),
so {w1 , m2 , m3 , t0v } 6= {a1 , b1 , a2 , b2 }.) We again apply Lemma 2.11 to the associated 4separation, to find that (Γ, X) ∈ O2 or (Γ, X) satisfies property (R1). Hence we may
assume m2 is not cofacial with t0v , and either S0 exists with s0 ∈ MP [z2 , m2 ), as desired, or
there is a path from P (t0v , s0 ] to MP (m2 , b1 ].
We now assume there is a path V from v 0 ∈ P (vt , s0 ] to v ∈ MP (m2 , b1 ]. Let Ov :=
P [a1 , t0v ] ∪ Tv ∪ T ∪ P [t0 , x1 ], from a1 to x1 . From our choice of m1 , we have a path Ow from
w1 to w2 in Bw which is disjoint from Ov . Let Ob := MP [b2 , w1 ] ∪ Ow ∪ MP [w2 , s1 ] ∪ S1 ∪
P [v 0 , s0 ] ∪ V ∪ MP [v, b1 ]. Now if we have a path from a2 to z2 in LP disjoint from Ob , then
together with Qx this shows x2 ∈ R(Ot ), contradicting the infeasibility of X in Γ. Hence
there can be no such path, and by planarity there must be a 2-cut in LP separating z2 from
a2 , with one vertex in MP [b2 , w1 ] ∪ MP [w2 , z1 ] and the other in MP [z2 , b1 ]. But such a 2-cut
also separates X 0 from a2 , and thus contradicts our choice of Z.
Claim 15. We may assume s0 = a1 .
Proof. Suppose s0 6= a1 . If s0 is cofacial with both z0 and z3 , we have a 4-separation (M, N )
of G with V (M ∩ N ) = {s0 , z3 , a2 , z0 } demonstrating that (Γ, X) satisfies (R1). So we may
assume that either S0 exists with s0 6= b1 , or T0 exists with t0 6= b2 .
Applying Claims 13 and 14, we have the following (Figure 43)
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Figure 44: Where s0 = a1
• 3-cut {h1 , h2 , h3 } in LP separating {b2 , a2 , b1 , s0 } from {w1 , w2 , z1 , z2 }
• 3-cut {m1 , m2 , m3 } in LP separating {t0 , b2 , a2 , b1 } from {ut , w2 , z1 , z2 }
• s00 ∈ P [a1 , t00 ), s0 ∈ MP [z2 , m2 ), and t0 ∈ MP (h1 , w1 ].
So we have z0 , h1 , t0 , m1 , w1 , z2 , h2 , s0 , m2 , z3 appearing in order along MP .
We choose the cut {m1 , m2 , m3 } to minimize MP [b2 , m1 ], and then to minimize MP [z2 , m2 ].
Observe from the planarity of LP that one of the following must hold:
(i) {m1 , h2 } is a 2-cut in LP
(ii) there is some v ∈ {h3 , m3 } such that {m1 , h2 , v} is a 3-cut in LP .
But (i) contradicts the choice of Z, and (ii) contradicts the choice of m2 .
Final Case. We now assume s0 = a1 .
From our choice of T , we find that either z0 and a1 share a finite face, or z0 = b2 . If
z0 6= b2 , we have a 4-separation (M, N ) of G with V (M ∩ N ) = {z0 , a1 , b1 , a2 }, such that
{a1 , b1 , a2 , b1 } ⊆ V (M ) and X ⊆ V (M ), and (Γ, X) satisfies (R1). Hence we may assume
that z0 = b2 . Since we assume there is no edge a1 b2 , we find T ⊆ C. Similarly, we may
89
assume z3 = b1 ; otherwise we find a 4-separation (M 0 , N 0 ) with V (M 0 ∩N 0 ) = {a1 , z3 , a2 , b2 }
demonstrating that (Γ, X) satisfies (R1).
Recall the path Us from u0s ∈ P [s0 , r 0 ] to us ∈ MP (z1 , s], chosen to minimize MP [z1 , us ],
and subject to this, minimize P [u0s , x1 ]. If u0s = s0 , we find that each of {w1 , w2 , s0 } and
{z1 , z2 , s0 } is a 3-cut in G, and (Γ, X) ∈ O3 . So we may assume that u0s 6= s0 .
Suppose u0s ∈ P (t0 , r 0 ]. Let O := P [a1 , t0 ] ∪ T ∪ MP [t, ut ] ∪ Ut , from a1 to x1 (Figure
44(a)). Then R ∪ Us ⊆ LO . Since x2 ∈
/ R(O), there must be some 2-cut V = {v1 , v2 } in LO
separating {b1 , a2 , b2 } from {us , z1 , w2 }. We may assume v2 ∈ MP [z2 , b1 ]. From our choice
of the 2-cut Z, we may assume v1 ∈ MO (w1 , w2 ). Thus we have a 4-separation (M, N ) of
G with V (M ∩ N ) = {v1 , v2 , s0 , w1 }, such that {a1 , b1 , a2 , b2 } ⊆ V (M ) and X ⊆ V (N ), and
(Γ, X) satisfies property (R1).
We may now assume u0s ∈ P (s0 , t0 ]. By our choice of Us , we have {w1 , w2 , t0 } is a 3-cut
in G (Figure 44(b)).
Suppose z2 6= b1 and there is some path Σ from a1 to σ ∈ MP (us , b1 ). Define Oσ :=
Σ ∪ MP [z2 , σ] ∪ Qx . Note that Oσ is disjoint from C[b1 , b2 ].
If there are disjoint paths from a2 to w2 and from b1 to z1 , each disjoint from MP [b2 , w1 ]∪
MP [z2 , σ], then X is feasible in Γ. Hence no such paths may exist, so we must have
some separation (M, N ) of LP of order at most three with {b1 , a2 , a1 } ⊆ V (M ) and
{z2 , z1 , w2 , w1 } ⊆ V (N ). If σ ∈ MP (z2 , b1 ), then σ ∈ V (M ).
From our choice of Z, we may assume this separation is of order three, and label V (M ∩
N ) = {h1 , h2 , h3 }, with h1 ∈ MP [b2 , w1 ] and h3 ∈ MP [z2 , σ]. But now we have a separation
(M 0 , N 0 ) of G with V (M 0 ∩ N 0 ) = {h1 , h2 , h3 , s0 }, such that {a1 , b1 , a2 , b2 } ⊆ V (M 0 ) and
X 0 ⊆ V (N 0 ), and we find that (Γ, X) satisfies property (R1).
Now suppose z2 6= b1 , and there is no path from a1 to MP (us , b1 ). Since we assume
there is no edge a1 b1 , we have us ∈ V (C) and us = s. So {z1 , z2 , s} is a 3-cut in G. We now
have a facial cycle D = P [u0s , r 0 ] ∪ R ∪ MP [r, us ] ∪ S, and separations (H1 , M1 ), (H2 , M2 ),
and (H3 , M3 ) with
1. V (H1 ∩ M1 ) = {us , z1 , z2 }, {a1 , a2 , b1 , b2 } ⊆ V (H1 ),
90
2. V (H2 ∩ M2 ) = {t0 , w2 , w1 }, {a1 , a2 , b1 , b2 } ⊆ V (H2 ),
3. V (H3 ∩ M3 ) = {t0 , u0s }, a1 ∈ V (M3 ), {a2 , b1 , b2 } ∈ V (H3 ),
4. X ⊆ V (M1 ∪ M2 ∪ D ∪ C[b2 , w1 ] ∪ C[us , b1 ]), X ∩ M1 6= ∅, and X ∩ M2 6= ∅.
So we have (Γ, X) ∈ O5 .
Finally, consider the case that z2 = b1 . If us = s, we have separations as above, and
(Γ, X) ∈ O5 . Let us assume, then, that us 6= s. Let Z 0 = Z 0 (Us ) be labeled as {z10 , z20 }. Then
we have an internal separation (Hz , Mz ) with Hz ∩ Mz = {z1 , z10 , b1 } (Figure 44(c)). Let
H1 = Hz and M1 = Mz , and let (H2 , M2 ) and (H3 , M3 ) be the separations associated with
the cuts {t0 , w2 , w1 } and {t0 , z10 , b1 }. These separations again demonstrate that (Γ, X) ∈ O5 .
This completes the proof of Lemma 3.14.
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CHAPTER IV
SPECIAL CASE |X| = 2
In this chapter, we address a limited class of disk graphs with |X| = 2. We prove a modified
version of our main theorem, which will later be applied in the proof of the general case.
4.1
Obstructions
For convenience, we describe the obstructions Θ1 . . . Θ6 when |X| = 2. Note that for each
i = 1, . . . , 6, we have Θi ⊆ Oi .
Definition 4.1. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C, and x1 , x2 ∈
V (G) \ {a1 , b1 , a2 , b2 }. We say (Γ, {x1 , x2 }) ∈ Θ1 if x1 and x2 are incident with a common
finite face, or {x1 , x2 } ⊆ V (C[b1 , b2 ]), or {x1 , x2 } ⊆ V (C[b2 , b1 ]).
Definition 4.2. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph, and x1 , x2 ∈ V (G)\{a1 , b1 , a2 , b2 }.
We say (Γ, {x1 , x2 }) ∈ Θ2 if G has a separation (H, K) of order 3 such that {x1 , x2 } ⊆ V (K),
and {a1 , a2 , b1 , b2 } ⊆ V (H).
Definition 4.3. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph, and x1 , x2 ∈ V (G)\{a1 , b1 , a2 , b2 }.
We say (Γ, {x1 , x2 }) ∈ Θ3 if there is some i ∈ {1, 2} such that for each j ∈ {1, 2}, one of
the following holds:
(i) xj ∈ C[b3−i , bi ], or
(ii) there is a finite face of G incident with both ai and xj , or
(iii) G has a separation (H, K) of order 3 such that ai ∈ V (H ∩ K), xj ∈ V (K),
{a1 , a2 , b1 , b2 } ⊆ V (H)
Definition 4.4. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C, and let
x1 , x2 ∈ V (G) \ {a1 , b1 , a2 , b2 }. We say (Γ, {x1 , x2 }) ∈ Θ4 if ∃i, j ∈ {1, 2}, xj ∈ V (C),
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and Ca = C[ai , xj ] if xj ∈ C[ai , bi ], Ca = C[xj , ai ] if xj ∈ C[b3−i , ai ] such that one of the
following holds:
1. ∃ a finite face of G incident with both x3−j and some vertex in V (Ca ); or
2. G has a separation (H, K) of order at most 4 with {xj , a1 , b1 , a2 , b2 } ⊆ V (H), x3−j ∈
V (K) and |V (H ∩ K) − V (Ca )| ≤ 2.
Definition 4.5. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C, and x1 , x2 ∈
V (G) \ {a1 , b1 , a2 , b2 }. We say (Γ, {x1 , x2 }) ∈ Θ5 if G has separations (H1 , M1 ), (H2 , M2 ),
and (H3 , M3 ), and a cycle D bounding a finite face of G such that the following hold for
some i, j ∈ {1, 2}:
1. V (H1 ∩ M1 ) = {u1 , v1 , w1 }, w1 ∈ V (C), {u1 , v1 } ⊆ V (D), {a1 , a2 , b1 , b2 } ⊆ V (H1 ),
2. V (H2 ∩ M2 ) = {u2 , v2 , w2 }, w2 ∈ V (C), {u2 , v2 } ⊆ V (D), {a1 , a2 , b1 , b2 } ⊆ V (H2 ),
3. V (H3 ∩ M3 ) = {u1 , u2 } ∪ ({w1 , w2 } ∩ {b1 , b2 }), ai ∈ V (M3 ), {a3−i , b1 , b2 } ∈ V (H3 ),
4. X ⊆ V (M1 ∪ M2 ∪ D ∪ C[b3−i , w1 ] ∪ C[w2 , bi ]), xj ∈ V (M1 ), and x3−j ∈ V (M2 ).
Definition 4.6. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C, and let
X ⊆ V (G) \ {a1 , b1 , a2 , b2 } with |X| ≥ 2. We say (Γ, X) ∈ Θ6 if G has separations (H1 , M1 )
and (H2 , M2 ) and a cycle D bounding a finite face of G such that the following hold for
some i ∈ {1, 2}:
1. V (D ∩ C) = {u1 , u2 }
2. For each j ∈ {1, 2} there is a path Cj ⊆ C from cj to {b1 , b2 } with V (Cj )∩{a1 , a2 } = ∅
3. V (Hj ∩ Mj ) = {uj , vj , wj }, vj ∈ V (D), w1 ∈ V (Cj ), {a1 , b1 , a2 , b2 } ⊆ V (Hj ), and
4. X ⊆ V (M1 ∪ M2 ∪ D ∪ C1 ∪ C2 )
4.2
Theorem 2
The results we state here are nearly identical to those of Theorem 1. Our only adjustments
come in the consideration of obstruction class O7 , and (Γ, X) satisfying reduction property
(R2).
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If (Γ, X) ∈ O7 and |X| < 3, we have (Γ, X) ∈ Θ1 ∪ Θ2 or (Γ, X) satisfies (R1).
Recall for disk graph Γ = (G, a1 , b1 , a2 , b2 ) and vertex set X with (Γ, X) satisfying
(R2), X can be partitioned into X1 = X ∩ V (C(b3−i , ai )), X2 = X ∩ V (C(ai , bi )), and
X3 = X \ (X1 ∪ X2 ) 6= ∅, and G has a separation (M, K) of order at most 4 such that one
of the following holds:
(i) X1 , X2 are nonempty, X1 ∪ X2 ∪ {a1 , b1 , a2 , b2 } ⊆ V (H), X3 ⊆ V (K), and |V (H ∩
K − C[b3−i , bi ])| ≤ 1
(ii) X2 = ∅, X1 ∪ {a1 , b1 , a2 , b2 } ⊆ V (H), X3 ⊆ V (K), and |V (H ∩ K − C[b3−i , ai ])| ≤ 2
(iv) X1 = ∅, X2 ∪ {a1 , b1 , a2 , b2 } ⊆ V (H), X3 ⊆ V (K), and |V (H ∩ K − C[ai , bi ])| ≤ 2
Suppose X = {x1 , x2 }. Then in case (i), we have (Γ, {x1 , x2 }) ∈ Θ1 ; in cases (ii) and (iii),
we have (Γ, {x1 , x2 }) ∈ Θ4 .
Theorem 2. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph, and let x1 , x2 ∈ V (G)\{a1 , b1 , a2 , b2 }.
Then one of the following holds.
(C1) {x1 , x2 } is feasible in Γ, or
(C2) (Γ, {x1 , x2 }) satisfies property (R1), or
(C3) (Γ, {x1 , x2 }) ∈
4.3
S6
i=1 Θi .
A special case
We will approach the proof of Theorem 2 using maximizing paths, and the lemmas of the
previous chapter. First, however, we deal with the special case when some vertex xi lies on
the outer cycle of G.
Lemma 4.1. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph. Let x1 , x2 ∈ V (G) \ {a1 , b1 , a2 , b2 },
with x1 6= x2 , and {x1 , x2 } infeasible in Γ. Suppose {x1 , x2 }∩V (C) 6= ∅. Then (Γ, {x1 , x2 }) ∈
Θ1 ∪ Θ4 .
Proof. We assume without loss of generality that x1 ∈ C(b2 , a1 ). If x2 ∈ C(b2 , b1 ), then
(Γ, {x1 , x2 }) ∈ Θ1 , so we may further assume that x2 ∈
/ C(b2 , b1 ).
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Figure 45: x1 ∈ V (C),
(Γ, {x1 , x2 }) ∈ Θ4
Let P := C[x1 , a1 ], and consider the graph H = G − P . Note that H is connected;
otherwise, there is some 2-cut of G contained in P , which contradicts (3, {a1 , b1 , a2 , b2 })connectivity of G.
If x2 is a cutvertex of H, then x2 is cofacial with some vertex of P , and (Γ, {x1 , x2 }) ∈ Θ4
with i = j = 1, Ca = P . Hence we may assume that there is a unique block Bx of H for
which x2 ∈ V (Bx ).
Suppose first that x2 ∈
/ V (LP ). Let B be the (LP ∪ P )-bridge of G which includes x2 ,
and let {v} = V (B ∩ LP ). Let ux , ua be the elements of V (B ∩ P ) which are closest to (and
possibly equal to) x1 and a1 respectively. Since G is (3, {a1 , b1 , a2 , b2 })-connected, ux 6= ua .
Then G has a separation (M, K) with V (M ∩ K) = {ux , ua , v} with {x1 , a1 , b1 , a2 , b2 } ⊆
V (M ) and x2 ∈ V (K), and we see that (Γ, {x1 , x2 }) ∈ Θ4 with i = j = 1, Ca = P .
Now suppose x2 ∈ V (LP ). (See Figure 45.) When S is a subgraph of LP , we call a
vertex v ∈ P a neighbor of S if there is a path from some s ∈ V (S) to v in G which is
internally disjoint from P ∪ LP .
First consider the case that either a2 ∈
/ Bx or a2 ∈ Bx is a cutvertex of H. Then
there is some block Ba 6= Bx of H with a2 ∈ V (Ba ) and Ba ⊆ LP . Let v ∈ V (Ba ) be
a cutvertex in H separating Bx , Ba (where v = a2 is possible). If Bx and Ba occur in
clockwise order along C ∩ LP , then let u be the neighbor of Bx closest to x1 on P , and
let T = {u, v, a1 , b1 }. Otherwise, let u be the neighbor of Bx closest to a1 on P , and let
T = {u, v, x1 , b2 }. In either case, we see that G has a separation (M, K) with M ∩ K = T
and {x1 , a1 , a2 , b1 , b2 } ⊆ V (M ), x2 ∈ V (K), and hence (Γ, {x1 , x2 }) ∈ Θ4 with i = j = 1
and Ca = P .
95
We now assume a2 ∈ V (Bx ) and a2 is not a cutvertex of H. Let D be the outer cycle
of the block Bx . Let v1 , v2 ∈ V (C) ∩ V (D) such that C[v1 , v2 ] = D[v1 , v2 ] is maximal. We
see that if x2 ∈ V (D[v2 , v1 ]), then x2 is cofacial with a vertex of P and (Γ, {x1 , x2 }) ∈ Θ4
with i = j = 1 and Ca = P . Let us assume, then, that x2 ∈
/ V (D[v2 , v1 ]).
Let Q = (C[b1 , b2 ] − C(v1 , v2 )) ∪ D[v2 , v1 ], so D[v2 , v1 ] ⊆ Q ⊆ (C ∪ D). Then Bx − Q
has no x2 to a2 path, since such a path along with P and Q would show that {x1 , x2 }
is feasible in Γ. So H has a separation (H1 , H2 ) of order 2 with V (H1 ∩ H2 ) ⊆ V (Q),
a2 ∈ V (H1 ), x2 ∈ V (H2 ).
Let ux be the neighbor of H2 closest to x1 on P , and ua the neighbor of H2 closest
to a1 on P . Then G has a separation (M, K) with V (M ∩ K) = V (H1 ∩ H2 ) ∪ {ux , ua },
{x1 , a1 , a2 , b1 , b2 } ⊆ V (M ), and x2 ∈ V (H2 ) ⊆ V (K). Hence (Γ, {x1 , x2 }) ∈ Θ4 with
i = j = 1 and Ca = P .
4.4
Proof of Theorem 2
Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C, and let x1 , x2 ∈ V (G) \
{a1 , b1 , a2 , b2 }. We wish to prove that one of the following holds:
(C1) {x1 , x2 } is feasible in Γ, or
(C2) (Γ, {x1 , x2 }) satisfies property (R1), or
(C3) (Γ, {x1 , x2 }) ∈
S5
i=1 Θi .
Proof. We assume that Γ does not satisfy (C1) or (C2), and show that (Γ, {x1 , x2 }) ∈
S5
i=1 Θi .
From Lemma 4.1, if {x1 , x2 } ∩ V (C) 6= ∅, then (Γ, {x1 , x2 }) ∈ Θ1 ∪ Θ4 . We may assume,
then, that {x1 , x2 } ∩ V (C) = ∅.
Recall that P(Γ,{x1 ,x2 }) is defined as the collection of all paths P from ai ∈ {a1 , a2 } to
xj ∈ {x1 , x2 } in G − (C[bi , b3−i ] ∪ {x3−j }) such that a3−i is not a cutvertex of LP . From
Lemma 3.1, either P(Γ,{x1 ,x2 }) 6= ∅ or (Γ, {x1 , x2 }) ∈ Θ3 , so we may assume that there is a
maximizing path P ∈ P(Γ,{x1 ,x2 }) . We further assume without loss of generality that the
path P is from a1 to x1 .
96
Now suppose x2 ∈
/ V (LP ). From Lemma 3.5, we find that (Γ, {x1 , x2 }) ∈ O1 ∩ O2 , so
(Γ, {x1 , x2 }) ∈ Θ1 ∩ Θ2 . Hence we may assume that x2 ∈ V (LP ). Applying Lemma 3.7, we
find that either (Γ, {x1 , x2 }) ∈ Θ2 ∪ Θ5 or x2 ∈ A(P ), so we assume that x2 ∈ A(P ). We
now apply Lemma 3.12 to see that if x2 ∈ MP then (Γ, {x1 , x2 }) ∈ Θ1 ∪ Θ2 ∪ Θ3 or satisfies
(R1), so we may assume x2 ∈
/ MP .
If there is some 2-separation (H, K) of LP with x2 ∈ V (H) \ V (K) and {b1 , b2 , a2 } ∈ K,
then by Lemma 3.14, (Γ, {x1 , x2 }) ∈ Θ1 ∪Θ2 ∪Θ3 . We may assume that no such 2-separation
exists. Together with x2 ∈
/ MP , this shows that there must be some path P2 from a2 to x2
in LP − MP , and x2 ∈ R(P ). Then MP provides a path from b1 to b2 disjoint from P and
P2 , and we find that {x1 , x2 } is feasible in Γ, a contradiction.
4.5
1-connected case
While we have considered graphs which are 2-connected, a slight modification gives similar
results for graphs which are connected, but not 2-connected.
Definition 4.7. Let G be a graph and let a1 , b1 , a2 , b2 be distinct vertices of G. We call
Γ = (G, a1 , b1 , a2 , b2 ) a weak disk graph if G is a connected graph drawn in a disk in the
plane with no pair of edges crossing such that a1 , b1 , a2 , b2 ∈ V (G) appear in clockwise order
along the boundary of the disk, and G is (3, {a1 , b1 , a2 , b2 })-connected.
Let Γ = (G, a1 , b1 , a2 , b2 ) be a weak disk graph, and let X ⊆ V (G) \ {a1 , b1 , a2 , b2 }. We
again say X is feasible in Γ if there exist three disjoint paths in G: from a1 to X, from a2
to X, and from b1 to b2 respectively. Otherwise, X is infeasible in Γ.
Note that the definitions of property (R1) and of Θ2 do not depend upon the 2connectivity of disk graphs, and hence can also be applied to weak disk graphs.
Lemma 4.2. Let Γ = (G, a1 , b1 , a2 , b2 ) be a weak disk graph such that G is not 2-connected.
Let x1 , x2 ∈ V (G) \ {a1 , b1 , a2 , b2 }. Then one of the following holds:
(1) {x1 , x2 } is feasible in Γ, or
(2) (Γ, {x1 , x2 }) satisfies property (R1), or
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Figure 46: Condition (4)
(3) (Γ, {x1 , x2 }) ∈ Θ2 , or
(4) G is a chain of at most three blocks such that
(a) each endblock of G contains exactly two vertices of {a1 , b1 , a2 , b2 }, and one of
{x1 , x2 },
(b) if G has three blocks, one of these is a single edge,
(c) there is a unique path P from b1 to b2 in ∂G − {a1 , a2 }, and
(d) {x1 , x2 } ∩ V (P ) 6= ∅.
Proof. Let v be a cutvertex of G. Since G is (3, {a1 , b1 , a2 , b2 })-connected, each component
of G − v must contain at least one of {a1 , b1 , a2 , b2 }. Label the components of G − v as
H1 , . . . , Hk where k ∈ {2, 3, 4}, and label the associated v-bridges of G as B1 , . . . , Bk .
We first claim that we may assume each component of G − v contains at least two
vertices of {a1 , b1 , a2 , b2 }. To see this, suppose there is some component Hj with Hj ∩
{a1 , b1 , a2 , b2 } = {u}.
Either {v, u} is a 2-cut in G, contradicting (3, {a1 , b1 , a2 , b2 })-
connectivity of weak disk graphs, or V (Hj ) = {u} and Bj is a single edge uv. But then we
have a 4-separation (H, K) of G with V (H ∩ K) = {v, a1 , b1 , a2 , b2 } \ {u}, {a1 , b1 , a2 , b2 } ⊆
V (H), and {x1 , x2 } ⊆ V (K), and (Γ, {x1 , x2 }) satisfies property (R1). Hence we may assume each component of G − v contains exactly two vertices of {a1 , b1 , a2 , b2 }, and G − v
has exactly two components, H1 and H2 .
From planarity and the order of {a1 , b1 , a2 , b2 } along the boundary of our embedding
disk, we note that for i ∈ {1, 2} we cannot have {a1 , a2 } ⊆ V (Hi ) and {b1 , b2 } ⊆ V (H3−i ).
We assume without loss of generality that {ai , bi } ⊆ V (Hi ).
98
If there is a cutvertex u separating ai from bi in Bi , then u is also a cutvertex of G, and
we apply the arguments above (for cutvertex v) to u. So we may assume that ai , bi appear
in a single block of Bi . If u is a cutvertex of Bi separating {ai , bi } from v, then we must
have the edge uv ∈ E(G); otherwise, {u, v} is a 2-cut in G violating (3, {a1 , b1 , a2 , b2 })connectivity of G. So we see that Bi is either 2-connected, or the union of a 2-connected
block and an edge uv. If each of B1 , B2 has a cutvertex, say u1 and u2 , then {u1 , u2 } is a
2-cut in G separating v from {a1 , b1 , a2 , b2 }, violating the (3, {a1 , b1 , a2 , b2 })-connectivity of
G. So at least one of B1 , B2 is 2-connected. (See Figure 46.)
We claim that each component of G − v must contain at least one vertex of {x1 , x2 }.
To see this, suppose there is some component Hi for which V (Hi ) ∩ {x1 , x2 } = ∅. Letting
H = Bi ∪ {a3−i , b3−i } and K = B3−i , we note that V (H ∩ K) = {v, a3−i , b3−i }. So we have
a separation (H, K) of G of order 3 such that {x1 , x2 } ⊆ V (K), and {a1 , a2 , b1 , b2 } ⊆ V (H),
and we see (Γ, {x1 , x2 }) ∈ Θ2 . So we assume without loss of generality that x1 ∈ H1 and
x2 ∈ H2 .
Now for each i ∈ {1, 2}, let Qi be the path in Bi from bi to v along ∂Bi , chosen so that
ai ∈
/ V (Qi ). Suppose there are paths Pi from ai to xi disjoint from Qi in each bridge Bi .
Then that paths P1 , P2 and Q1 ∪ Q2 would show that {x1 , x2 } is feasible in Γ. Hence no
such paths may exist. So some xi must either lie on the path Pi , or be separated from ai
by a 2-cut Ti ⊆ V (Qi ). But such a 2-cut Ti would violate (3, {a1 , b1 , a2 , b2 })-connectivity of
G, so we find xi ∈ Qi , and condition (4) of this lemma is satisfied.
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CHAPTER V
PROOF OF GENERAL CASE
We now extend Theorem 2 to the more general case |X| ≥ 2. We work inductively on |X|,
making use of maximizing paths and the lemmas of previous sections.
We begin with two cases which can be handled without the use of either maximizing
paths or induction. These use direct definitions of path routings. We then handle the
situation that some vertex of X is on the outer cycle of G. This makes use of the results
where |X| = 2, considering the relative positions of vertex pairs in X. It also employs
Theorem 1 inductively.
We then establish several more technical lemmas, and handle several cases by considering
maximizing paths. The remaining possibilities are dispatched by considering the obstructive
relationship within subsets of X, and their possible combinations, again employing Theorem
1 inductively.
We combine these individual results to prove Theorem 1.
5.1
Independent inductive cases
We first consider the cases where removal of one vertex from the set X leaves us with a disk
graph in O5 or O6 . We approach this by considering path routings which make use of the
identified vertex.
Lemma 5.1. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C, and let X =
(X 0 ∪ {x1 }) ⊆ V (G) \ {a1 , b1 , a2 , b2 } with |X 0 | ≥ 2, and X infeasible in Γ. If (Γ, X 0 ) ∈ O5 ,
then either (Γ, X) satisfies property (R1) or (Γ, X) ∈ O2 ∪ O4 ∪ O5 .
Proof. Assume (Γ, X 0 ) satisfies the definition of O5 with i = 1, so G has separations
(H1 , M1 ), (H2 , M2 ), and (H3 , M3 ), and a cycle D bounding a finite face of G such that
the following hold:
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Figure 47: Subgraph definitions for (Γ, X 0 ) ∈ O5
1. V (H1 ∩ M1 ) = {u1 , v1 , w1 }, w1 ∈ V (C), {u1 , v1 } ⊆ V (D), {a1 , a2 , b1 , b2 } ⊆ V (H1 ),
2. V (H2 ∩ M2 ) = {u2 , v2 , w2 }, w2 ∈ V (C), {u2 , v2 } ⊆ V (D), {a1 , a2 , b1 , b2 } ⊆ V (H2 ),
3. V (H3 ∩ M3 ) = {u1 , u2 } ∪ ({w1 , w2 } ∩ {b1 , b2 }), a1 ∈ V (M3 ), {a2 , b1 , b2 } ∈ V (H3 ),
4. if w1 ∈
/ {b1 , b2 }, then u1 ∈ V (C), and if w2 ∈
/ {b1 , b2 }, then u2 ∈ V (C), and
5. X 0 ⊆ V (M1 ∪ M2 ∪ D ∪ C[b2 , w1 ] ∪ C[w2 , b1 ]).
Assume further that these separations are chosen with M1 and M2 each maximal. See
Figure 47 for illustration of the following definitions. Let Go = M1 ∪ M2 ∪ D ∪ C[b2 , w1 ] ∪
C[w2 , b1 ], so X 0 ⊆ V (G0 ). Let Ui = Hi ∩ Mi for i ∈ {1, 2, 3}. Let H = H1 ∩ H2 ∩ H3 , and
note that G has exactly one H-bridge, with attachments {w1 , v1 , v2 , w2 }. Call this bridge
B. We may assume x1 ∈
/ V (Go ), or we have (Γ, X) ∈ O5 . Hence x1 ∈ M3 ∪ H.
For i = 1, 2, let Ei be the facial cycle of G including wi and vi , such that E1 [v1 , w1 ] ⊆ H1 ,
E1 [w1 , v1 ] ⊆ M1 , E2 [v2 , w2 ] ⊆ M2 , and E2 [w2 , v2 ] ⊆ H2 .
Case 1
Suppose x1 ∈ V (H).
We claim that we may assume there is some path Q2 from a2 to x1 in H such that Q2
is disjoint from C[b2 , w1 ] ∪ D[v1 , v2 ] ∪ E2 [w2 , v2 ]. (See Figure 48.)
Let Q := C[b2 , w1 ] ∪ E1 [v1 , w1 ] ∪ D[v2 , v1 ] ∪ E2 [w2 , v2 ] ∪ C[w1 , b1 ]. Since V (E1 ∩ E2 ) ⊆
V (D), and x1 ∈
/ V (D), we have x1 ∈
/ V (E1 ) or x1 ∈
/ V (E2 ).
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(b) x2 ∈ D[v2 , v1 )
(c) x2 ∈ C[w1 , b1 )
Figure 48: Case 1, x1 ∈ V (H)
Suppose some 2-cut T = {t1 , t2 } of H with T ⊆ V (Q) separates x1 from a2 . Note that
since G is (3, {a1 , b1 , a2 , b2 })-connected, Q[t1 , t2 ] must include at least one of {w1 , v1 , v2 , w2 }.
If we have tj ∈ V (C) and t3−j ∈ V (D) for some j ∈ {1, 2}, then T violates the maximality
of M1 and M2 . If T ⊆ V (Q ∩ C), we have a 3-cut T ∪ {a1 } demonstrating that (Γ, X) ∈ O2 .
So if such a 2-cut T exists, we may assume T ∩ V (E1 ∪ E2 ) 6= ∅. So there is some i ∈ {1, 2}
such that T ∩ V (Ei ) 6= ∅ and x1 ∈
/ V (E3−i ).
Choose T to be a maximal such 2-cut, and assume by symmetry that T ∩ V (E1 ) 6= ∅
and x1 ∈
/ V (E2 ). Let t1 ∈ T ∩ V (E1 ). Now there is a path Q2 from a2 to x1 in H which is
disjoint from Q[b2 , t1 ) ∪ Q[v1 , b1 ].
Let Q1 := C[b2 , w1 ]∪E1 [w1 , v1 ]∪Q[v1 , b1 ]. If there is some x2 ∈ X 0 such that x2 ∈
/ V (Q1 ),
102
then there is a path from a1 to x2 in G disjoint from each of Q1 and Q2 , and X is feasible in
Γ, a contradiction. So we must have X 0 ⊆ V (Q1 ). If X 0 ⊆ V (C[b2 , w1 ]∪E1 [w1 , v1 ]), we have
(Γ, X) ∈ O4 . We assume this is not the case, so there is some x2 ∈ X 0 ∩V (Q[v1 , b1 ]). Choose
such a vertex x2 to minimize Q1 [x2 , b1 ]. Since X 0 ⊆ V (M1 ∪ M2 ∪ D ∪ C[b2 , w1 ] ∪ C[w2 , b1 ]),
we have either x2 ∈ D[v2 , v1 ) or x2 ∈ C[w1 , b1 ).
Now {x1 , x2 } is feasible in Γ if there are disjoint paths from b2 to w1 , from a2 to x1 , and
from b1 to v1 in H − x2 . Hence no such paths may exist. So we must have some 3-separation
(H 0 , M 0 ) of H with {b2 , a2 , b1 } ⊆ V (H 0 ), {w1 , x1 , v1 } ⊆ V (M 0 ), and x2 ∈ V (H 0 ∩ M 0 ).
If x2 ∈ D[v2 , v1 ) (Figure 48(b)), then there is a 4-separation (G1 , G2 ) of G with V (G1 ∩
G2 ) = V (H 0 ∩ M 0 ) ∪ {u1 } and {a1 , b1 , a2 , b2 } ⊆ V (H 0 ). By our choice of x2 , we have
X ⊆ V (M 0 ). Hence (Γ, X) satisfies property (R1).
If x2 ∈ C[w1 , b1 ), then there is a 4-separation (G1 , G2 ) of G with V (G1 ∩ G2 ) = V (H 0 ∩
M 0 ) ∪ {a1 } and {a1 , b1 , a2 , b2 } ⊆ V (H 0 ). Again by our choice of x2 , we have X ⊆ V (M 0 ),
so (Γ, X) satisfies property (R1).
Case 2
Suppose x1 ∈ V (M3 ).
If |U3 | = 4, then V (H3 ∩M3 ) = {u1 , u2 , b1 , b2 }. Note that M3 and H are now symmetric,
so we simply apply the argument of Case 1 above.
(2.1)
Suppose |U3 | = 2.
Then we have u1 , u2 ∈ V (C ∩ D). We have assumed x1 ∈
/ V (Go ), so x1 ∈
/ V (D). Since
G is (3, {a1 , b1 , a2 , b2 })-connected, there can be no 2-cut T ⊆ V (D) separating x1 from a1 .
Hence there is a path P1 from a1 to x1 in H3 −V (D). Let P2 = C[b2 , u1 ]∪D[u1 , u2 ]∪C[u2 , b1 ],
and note P1 and P2 are disjoint.
If X 0 ⊆ V (P2 ), then (Γ, X) ∈ O4 , so we assume this is not the case. Let X 00 = X 0 \V (P2 ).
If there is some path P3 from a2 to X 00 in G − V (P2 ), then X is feasible in Γ, a
contradiction. Hence we may assume no such path exists. From planarity, we see that there
must be a 2-cut T ⊆ V (P2 ) in G separating X 00 from a2 . In fact, T ⊆ V (C ∩ P2 ), and we
find that T ∪ {a1 } is a 3-cut in G separating X 00 from {a1 , b1 , a2 , b2 }. If X 00 = X 0 , we have
(Γ, X) ∈ O2 . If X 00 6= X 0 , we have (Γ, X) ∈ O4 .
103
(2.2)
Suppose |U3 | = 3.
We assume without loss of generality that w2 = b1 , so V (H3 ∩ M3 ) = {u1 , u2 , b1 }, as in
Figure 47. We claim that there is a path P1 from a1 to x1 in M3 − D. If not, then either
x1 ∈ V (D), or some cut T ⊆ V (D[u1 , u2 ]), |T | ≤ 2, separates a1 from x1 in M3 . But we have
assumed x1 ∈
/ V (D), and any such cutset T violates the (3, {a1 , b1 , a2 , b2 })-connectivity of
G. Let Q be any path from u2 to b1 in M2 , and let P2 = C[b2 , u1 ] ∪ D[u1 , u2 ] ∪ Q, from b2
to b1 in G. Note that P1 and P2 are disjoint.
Notice that {u1 , v2 , b1 , a1 } is a 4-cut in G, leading to a separation (H 0 , M 0 ) with {a1 , b1 , a2 , b2 } ⊆
V (H 0 ) and x1 ∈ V (M 0 ). If X 0 ⊆ V (M 0 ), we find (Γ, X) satisifes (R1). If X 0 ⊆ V (M 0 ) ∪
C[b2 , u1 ], then (Γ, X) ∈ O4 . Hence we may assume X 0 ⊆ V (M 0 ) ∪ C[b2 , u1 ], so X 0 * V (P2 ).
Let X 00 = X 0 \ V (P2 ).
Now if there is a path P3 from a2 to X 00 disjoint from P2 , then X is feasible in Γ, a
contradiction. Hence no such path may exist. So by planarity there must be some 2-cut
T ⊆ V (P2 ) in H ∪ M1 separating X 00 from a2 . From planarity and (3, {a1 , b1 , a2 , b2 })connectivity of G, we see that T = {t1 , t2 }, with t1 ∈ C[b2 , w1 ] and t2 ∈ {v2 , b1 }. If
t2 = v2 , we have a 3-cut T ∪ {u1 } in G, violating the maximality of M1 . Hence we assume
t2 = b1 , and we have a separation (H0 , M0 ) of G with V (H0 ∩ M0 ) = T ∪ {a1 }, such that
{a1 , b1 , a2 , b2 } ⊆ V (H0 ) and X 00 ∪ {x1 } ⊆ V (M0 ). If X ⊆ V (M0 ), then (Γ, X) ∈ O2 .
Otherwise, X ⊆ V (M0 ∪ C[b2 , t1 ], and (Γ, X) ∈ O4 .
Lemma 5.2. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C, and let X =
(X 0 ∪ {x1 }) ⊆ V (G) \ {a1 , b1 , a2 , b2 } with |X 0 | ≥ 2, and X infeasible in Γ. If (Γ, X 0 ) ∈ O6
and X 0 * V (C), then either (Γ, X) is reducible or (Γ, X) ∈ O6 .
Proof. We may assume (Γ, X 0 ) satisfies the definition for O6 with i = 1, as shown in Figure
49. That is, G has separations (H1 , M1 ) and (H2 , M2 ) and a cycle D bounding a finite face
of G such that:
1. V (H1 ∩M1 ) = {u1 , v1 , w1 }, u1 ∈ V (C∩D), v1 ∈ V (D), w1 ∈ C[u1 , b2 ], {a1 , a2 , b1 , b2 } ⊆
V (H1 ),
104
2. V (H2 ∩M2 ) = {u2 , v2 , w2 }, u2 ∈ V (C∩D), v2 ∈ V (D), w2 ∈ C[u2 , b1 ], {a1 , a2 , b1 , b2 } ⊆
V (H2 ),
3. X 0 ⊆ V (M1 ∪ M2 ∪ D ∪ C[w1 , b2 ] ∪ C[w2 , b1 ])
Assume these separations are chosen to maximize M1 and M2 . We may assume x1 ∈
/
V (M1 ∪ M2 ∪ D ∪ C[w1 , b2 ] ∪ C[w2 , b1 ]); otherwise we have (Γ, X) ∈ O6 . Notice that
{w1 , v1 , u2 } is a 3-cut in G. Let H be the {w1 , v1 , u2 }-bridge of G which contains {a1 , b2 }.
Let M be the {w1 , v1 , u2 }-bridge containing M1 and M2 . By symmetry, we may assume
x1 ∈ V (H) \ V (C[w1 , b2 ] ∪ D[v1 , v2 ]).
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Figure 49: (Γ, X 0 ) ∈ O6
We claim first that we may assume there is a path P1 from a1 to x1 in H − V (C[w1 , b2 ] ∪
D). To see this, suppose not. Since we assume x1 ∈
/ V (C[w1 , b2 ] ∪ D), there must be some
cut T in H with |T | ≤ 2 separating a1 from x1 . Since G is (3, {a1 , b1 , a2 , b2 })-connected, we
must have T = {t1 , t2 } with t1 ∈ C[w1 , b2 ] and t2 ∈ V (D). But this violates the maximality
of M1 . So no such cutset exists, and we have a path P1 .
Let Q := C[w1 , b2 ] ∪ ∂M1 [w1 , v1 ] ∪ D[v1 , u2 ] ∪ C[u2 , b1 ], and note that Q is a path from
b1 to b2 which is disjoint from P1 .
Now if there is a path P2 from a2 to X in M , disjoint from Q, then we would have
X feasible in Γ. So there can be no such path, and for each x ∈ X ∩ V (M ), either
x ∈ V (Q) or there is a 2-cut T in M separating a2 from x, with T ⊆ V (Q). From
(3, {a1 , b1 , a2 , b2 })-connectivity and planarity of G, we find that so such 2-cut exists, so we
105
must have X ∩ V (M ) ⊆ V (Q). We find that there is a 4-separation (H 0 , M 0 ) of G with
V (H 0 ∩ M 0 ) = {a1 , u1 , u2 , b2 }, such that {a1 , b1 , a2 , b2 } ⊆ V (H 0 ) and X ⊂ V (M 0 ) ∪ C[u2 , b1 ].
Since X 0 * V (C), we find |X \ V (C[u2 , b1 ])| ≤ 2. If X ⊆ V (M 0 ), then (Γ, X) satisfies
property (R1). Otherwise, (Γ, X) satisfies property (R2).
5.2
Vertex of X on outer face
Several obstruction classes require some vertex of X on the outer face of G. Where X
has only two vertices, one of which is on the outer cycle, we have seen that their relative
positions are extremely limited. We will make use of this by considering vertex pairs.
Remark 5.1. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C, and let X ⊆
V (G) \ {a1 , b1 , a2 , b2 } with |X| ≥ 3, and X infeasible in Γ. Suppose x1 ∈ V (C), and let
X 0 = X \ {x1 }. Each pair {x1 , x} with x ∈ X 0 must be infeasible in Γ. By an application
of Lemma 4.1, we find that (Γ, {x1 , x}) ∈ Θ1 ∪ Θ4 . Letting Ca = C[ai , x1 ] if x1 ∈ C[ai , bi ],
Ca = C[x1 , ai ] if x1 ∈ C[b3−i , ai ], one of the following holds:
(A1) ∃ a face of G incident with both x and some vertex in Ca , or
(A2) G has a separation (H, M ) of order at most 4 with {x1 , a1 , a2 , b1 , b2 } ⊆ V (H), x ∈
V (M ), and |V (H ∩ M ) − Ca | ≤ 2
We will use this fact in several of the following arguments.
Definition 5.1. Where D is a cycle in a plane graph G, we denote by Int(D) the open
disk bounded by D, and by Int(D) the closed disk bounded by D.
Lemma 5.3. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C, and let X ⊆
V (G) \ {a1 , b1 , a2 , b2 } with |X| ≥ 3, X infeasible in Γ, and some x1 ∈ X ∩ V (C). Let X 0 =
X \{x1 }. If there is a cycle D in G such that X 0 ⊆ Int(D) and {a1 , b1 , a2 , b2 }∩ Int(D) = ∅,
then (Γ, X) satisfies (R2) or (Γ, X) ∈ O4 .
Proof. Assume by symmetry that x1 ∈ C[b2 , a1 ], and hence Ca = C[x1 , a1 ]. Label X 0 =
{x2 , . . . , xn }. For any xk ∈ X 0 which satisfies (A2), choose a separation (H, M ) with M
maximal, and call it (Hk , Mk ).
106
Suppose (H, M ) is a separation of G of order at most 4 with {x1 , a1 , b1 , a2 , b2 } ⊆ V (H)
and |V (H ∩ M − Ca )| ≤ 2. If X 0 ⊆ V (M ), then we find (Γ, X) ∈ O4 . Hence we may assume
X 0 * V (M ), and therefore V (D) * V (M ).
Now suppose xk is not cofacial with any vertex of Ca . Then we have the separation
(Hk , Mk ), and xk ∈
/ V (Hk ∩ Mk ). Hence |V (D) ∩ V (Mk )| ≥ 2. But we assume V (D) *
V (Mk ), so V (Mk ∩ Kk ) is a cutset in G separating D. Furthermore, V (Hk ∩ Mk − Ca ) ⊆
V (D), so |V (Hk ∩ Mk − Ca )| = 2. Label V (Hk ∩ Mk − Ca ) = {vk , wk } such that D[vk , wk ] ⊆
Mk , and note that each of vk , wk is cofacial with a vertex of Ca . (See Figure 50.)
Recall our choice of each separation (Hk , Mk ) to maximize Mk . By planarity, we find
that there is a single separation (H, M ) with (H, M ) = (Hk , Mk ) for any xk not cofacial
with a vertex of Ca . Label V (H ∩ K) = {ux , ua , v, w} with v = vk , w = sk , ux , ua ∈ V (C)
chosen to minimize each of C[x1 , ux ] and C[ua , a1 ].
Now suppose there is some x ∈ X 0 with x ∈
/ V (M ). Then x is cofacial with some
vertex u ∈ V (Ca ). If u ∈ C[x1 , ux ], we have a separation (H 0 , M 0 ) of G with V (H 0 , M 0 ) =
{ua , w, x, u}, contradicting the maximality of M . Similarly, if u ∈ C[ua , a1 ], there is a
separation (H 0 , M 0 ) of G with V (H 0 , M 0 ) = {u, x, v, ux }, again contradicting the maximality
of M . So we find X 0 ⊆ V (Mu ), and (Γ, X) ∈ O4 or (Γ, X) satisfies (R2).
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Figure 50: x1 ∈ V (C), X 0 ⊂ Int(D).
Lemma 5.4. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C, and let X ⊆
V (G) \ {a1 , b1 , a2 , b2 } with |X| = n ≥ 3, X infeasible in Γ, and X ∩ V (C) 6= ∅. If Theorem
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1 holds for vertex sets of size n − 1, then either (Γ, X) is reducible, or (Γ, X 0 ) ∈
S7
i=1 Oi .
Proof. By symmetry, assume that X ∩ C(b2 , a1 ) 6= ∅. Define X1 = X ∩ C(b2 , a1 ), X2 =
X ∩ C(a1 , b1 ), and X3 = X \ V (C(b2 , b1 )). If X ⊆ C(b2 , b1 ), then (Γ, X) ∈ O1 , so we assume
that X3 6= ∅. Let x1 ∈ X1 be chosen to maximize C[x1 , a1 ], and let X 0 = X \ {x1 }.
We may assume X 0 is not feasible in Γ; otherwise X is feasible in Γ, a contradiction.
Now |X 0 | = n − 1, so we may apply Theorem 1 to find that either (Γ, X 0 ) is reducible or
S
(Γ, X 0 ) ∈ 7i=1 Oi .
If (Γ, X 0 ) ∈ O5 , we apply Lemma 5.1. If (Γ, X 0 ) ∈ O6 , apply Lemma 5.2. If (Γ, X 0 ) ∈ O1 ,
we have some facial cycle D in G with X 0 ⊆ V (D), and we apply Lemma 5.3. If (Γ, X 0 ) ∈ O2
or satisfies (R1), G has a separation (H, K) of order at most 4 such that X 0 ⊆ V (K),
{a1 , b1 , a2 , b2 } ⊆ V (H). Choose such a separation with K maximal. Let D be the facial
cycle of H such that V (H ∩ K) ⊆ V (D). (Note from the (3, {a1 , b1 , a2 , b2 })-connectivity of
G and maximality of K that this cycle is unique.) Apply Lemma 5.3.
We assume, then, that (Γ, X 0 ) ∈ O3 ∪ O4 ∪ O7 or satisfies property (R2).
Case 1
|X ∩ V (C1 )| ≥ 2 for some C1 ∈ {C[a1 , b1 ], C[b1 , a2 ], C[a2 , b2 ], C[b2 , a1 ]}.
The argument is symmetric, so we may assume |X1 | ≥ 2. By our choice of x1 , we have
X 0 ∩ C[x1 , a1 ] 6= ∅.
(1.1)
Suppose (Γ, X 0 ) ∈ O3 .
If (Γ, X 0 ) ∈ O3 with respect to a1 , then we find (Γ, X) ∈ O3 . Assume, then, that
(Γ, X 0 ) ∈ O3 with respect to a2 . Let x2 ∈ X 0 ∩ C[x1 , a1 ] be chosen to minimize C[x1 , x2 ].
Either x2 is cofacial with a2 , or G has a separation (H, K) of order 3 such that a2 ∈ V (H ∩
K), x2 ∈ V (K), {a1 , a2 , b1 , b2 } ⊆ V (H). In either case, there is some vertex v ∈ C[b2 , a1 ]
which is cofacial with a2 . Hence there is a separation (H1 , K1 ) of G with V (H1 ∩ K1 ) =
{v, a2 , b2 }, x1 ∈ V (K1 ), and {a1 , a2 , b1 , b2 } ⊆ V (H1 ). This shows (Γ, X) ∈ O3 , with respect
to a2 .
(1.2)
Suppose (Γ, X 0 ) ∈ O4 or satisfies (R2).
108
We have a partition X10 , X20 , X30 of X 0 , some i, j ∈ {1, 2}, x0 ∈ Xj0 , and Ca0 = C[ai , x0 ]
if x0 ∈ C[ai , bi ], Ca0 = C[x0 , ai ] if x0 ∈ C[b3−i , a1 ], and a face or separation of G satisfying
the requirements of O4 or property (R2). Since for each x ∈ (X1 \ {x1 }), y ∈ X3 , the pair
{x, y} must be infeasible, we may assume i = j = 1, and Ca0 = C[x0 , a1 ]. So X10 = X1 \ {x1 },
X20 = X2 , X30 = X3 . From our choice of x1 to maximize C[x1 , a1 ], we have Ca0 ⊆ Ca , and
the face or separation which demonstrates that (Γ, X 0 ) ∈ O4 or satisfying property (R2)
also demonstrates (Γ, X) ∈ O4 or satisfying (R2), respectively.
(1.3)
Suppose (Γ, X 0 ) ∈ O7 .
Let (H, K) be the separation and D the cycle of G satisfying the requirements of O7 .
That is, for some i ∈ {1, 2}, the following hold
1. V (H ∩ K) = {u, v, w}, w ∈ C[b3−i , bi ] − ai , and u, v ∈ V (D)
2. {a1 , a2 , b1 , b2 } ⊆ V (H), X ∩ V (K) * V (H),
3. If w ∈ C(b3−i , ai ), then u ∈ C(w, ai ] and X ⊆ V (K ∪ D ∪ C(b3−i , w])
4. If w ∈ C(ai , bi ), then u ∈ C[ai , w) and X ⊆ V (K ∪ D ∪ C[w, bi ))
5. If w = b3−i , then X ⊆ V (K ∪ D).
Since X 0 ∩C[b2 , a1 ] 6= ∅, we must have i = 1. From 3-5 above, u ∈ V (C), so (X1 \{x1 }) ⊆
V (C(b2 , u]). From our choice of x1 to maximize C[x1 , a1 ], we find that x1 ∈ V (C(b2 , u]),
and hence (Γ, X) ∈ O7 .
Case 2
X1 = {x1 } and X2 = {x2 }.
Label X3 = {x3 , . . . , xn }, and recall that for each x ∈ {x1 , x2 } and y = xk ∈ X3 we have
either a face satisfying (A1) above, or a separation of G satisfying (A2) above.
Claim 1. For each xk ∈ X3 , there is some separation (Ok , Nk ) of G of order at most 3 such
that {b1 , a2 , b2 , x1 , x2 } ⊆ V (Ok ), {xk , a1 } ⊆ V (Nk ), and |V (Ok ∩ Nk − C[b2 , b1 ])| ≤ 1.
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Proof of claim 1 First suppose xk is cofacial with some v1 ∈ C[x1 , a1 ] and v2 ∈ C[a1 , x2 ].
Then we have the desired separation with V (Ok ∩ Nk ) = {x, v1 , v2 }.
Now suppose xk is not cofacial with any vertex of C[x1 , a1 ].
Then there is some
separation (H, M ) of order at most 4 with {x1 , a1 , a2 , b1 , b2 } ⊆ V (H), xk ∈ V (M ), and
|V (H ∩ M ) − C[x1 , a1 ]| ≤ 2. Label V (H ∩ M ) = {cx , ca , m1 , m2 } in clockwise order along
the boundary of M , with cx , ca ∈ V (C), letting ca = cx or m1 = m2 as appropriate if the
separation is of order 3 (Figure 51).
Recall that we have assumed xk not cofacial with any vertex of C[x1 , a1 ]. If xk is cofacial
with some vertex v2 ∈ C[a1 , x2 ], we must have m2 cofacial with v2 (Figure 51(a)). Hence
there is a separation as desired with V (Ok ∩ Nk ) = {c1 , m2 , v2 }.
Let us assume, then, that m2 is not cofacial with any vertex of C[a1 , x2 ]. We must have
some separation (H 0 , M 0 ) of order at most 4 with {x2 , a1 , b1 , a2 , b2 } ⊆ V (H 0 ), xk ∈ V (M 0 ),
and |V (H 0 ∩ M 0 ) − C[a1 , x2 ]| ≤ 2. Label V (H 0 ∩ M 0 ) = {c0a , c0x , m01 , m02 } in clockwise order
along the boundary of M 0 , with c0a , c0x ∈ V (C), letting c0a = c0x or m01 = m02 as appropriate
if the separation is of order 3 (Figures 51(b), 51(c)).
By planarity, we may choose m02 ∈ C[b2 , ca ], and c0a = a1 , and we have the desired
separation with V (Ok , Nk ) = {c0x , m01 , m02 }. This proves Claim 1
Assume each separation (Ok , Nk ) is chosen with Nk maximal. If X3 ⊆ V (Ni ) for some
i ∈ {3, . . . , n}, then we see (Γ, X) satisfies property (R2). So we may assume this is not
the case. Hence there are i, j ∈ {3, . . . , n} such that xi ∈
/ V (Nj ), xj ∈
/ V (Ni ). Consider the
separations (Oi , Ni ) and (Oj , Nj ). Let V (Oi , Ni ) = {u1 , u2 , u3 } (or {u1 , u2 } if the separation
is of order 2), with u1 ∈ C[x1 , a1 ] and u2 ∈ C[a1 , x2 ]. Similarly, let V (Oj , Nj ) = {v1 , v2 , v3 }
(or {v1 , v2 } if the separation is of order 2), with v1 ∈ C[x1 , a1 ] and v2 ∈ C[a1 , x2 ]. From our
choice of i and j, we see that neither Nj ⊆ Ni nor Ni ⊆ Nj is true. Hence we may assume
that u1 , v1 , a1 , u2 , v2 appear in this clockwise order along C (with equality possible among
pairs in {v1 , a1 , u2 }).
Now from planarity, and maximality of Ni and Nj , there are separations (H1 , M1 ),
(H2 , M2 ), (H3 , M3 ) of G such that V (H1 ∩ M1 ) = {u1 , v1 , u3 }, V (H2 ∩ M2 ) = {u2 , v2 , v3 },
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c0a = a1
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(b) (A2), (A2) combination
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(c) (A2), (A2) combination
Figure 51: X1 = {x1 }, X2 = {x2 }, separation (Ok , Nk ).
V (H3 ∩M3 ) = {v1 , u2 }. Observe further that {b1 , a2 , b2 , x1 , x2 } ⊆ V (H1 ∩H2 ∩H3 ), xi ∈ M1 ,
and xj ∈ M2 (Figure 52).
Claim 2. We may assume X3 ⊆ V (M1 ∪ M2 ∪ M3 ).
Proof of claim 2 Suppose not. So there is some xk ∈ X3 with xk ∈
/ V (M1 ∪ M2 ∪ M3 ).
Consider the associated separation (Ok , Nk ), and let V (Ok , Nk ) = {w1 , w2 , w3 } (or {w1 , w2 }
if the separation is of order 2). From planarity we find that one of (w1 , u1 , v1 , a1 , w2 , u2 , v2 ),
(w1 , u1 , v1 , a1 , u2 , v2 , w2 ), or (u1 , v1 , w1 , a1 , u2 , v2 , w2 ) appear in clockwise order along C.
But then there is separation (H, M ) of G with V (H∩M ) ∈ { {w1 , w2 }, {w1 , u2 }, {w1 , u2 , w3 },
{v1 , w2 }, {v1 , w2 , w3 }}, contradicting the maximality of Ni or Nj . This proves Claim 2.
Claim 3. We may assume X3 ∩ M3 lies on the boundary of M3 .
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Figure 52: X1 = {x1 }, X2 = {x2 }, leading to (Γ, X) ∈ O5 .
Proof of Claim 3 Suppose we have some xk ∈ V (M3 ) which is not incident with the
infinite face of M3 . Since G is (3, {a1 , b1 , a2 , b2 })-connected, and by maximality of M1
and M2 , we see that M3 is 2-connected. Let D be the outer cycle of M3 . Again from
(3, {a1 , b1 , a2 , b2 })-connectivity of G, there is a path P from a1 to xk in M3 − D[u2 , v1 ].
Now let Pb = C[b2 , v1 ] ∪ D[u2 , v1 ] ∪ C[v1 , b1 ]. Note xi ∈
/ Pb . Suppose there is a vertex
set T ⊆ V (Pb ) separating xi from a2 . By planarity, we must have T = {t1 , t2 } with
t1 ∈ C[b2 , v1 ], t2 ∈ C[v1 , b1 ]. Hence there is a separation (H, M ) of G with V (H ∩ M ) = T ,
X3 ∪ {a1 } ⊆ V (M ), demonstrating (Γ, X) ∈ O4 . Hence we may assume no such separating
set exists, and we find xi ∈ R(P ), so X is feasible, a contradiction.
The separations (H1 , M1 ), (H2 , M2 ), and (H3 , M3 ) now demonstrate (Γ, X) ∈ O5 .
Case 3
X1 = {x1 } and X2 = ∅.
We may now assume by symmetry that |X3 ∩ V (C)| ≤ 1.
(3.1)
Suppose (Γ, X 0 ) ∈ O3 .
Then ∃ i ∈ {1, 2} such that for each x ∈ X 0 one of the following holds
(i) there is a finite face of G incident with both ai and x, or
(ii) G has a separation (H, K) of order 3 such that ai ∈ V (H∩K), x ∈ V (K), {a1 , a2 , b1 , b2 } ⊆
V (H), or
112
(iii) x ∈ C[b3−i , bi ].
If i = 1, we find that x1 ∈ C[b3−i , bi ], and (Γ, X) ∈ O3 . Hence we may assume i = 2.
Let Ca := C[x1 , a1 ].
Let M be the collection of all separations (H, M ) of G with V (H ∩ M ) = {u, v, a2 , b2 },
such that u ∈ V (Ca ), {u, v} 6= {a1 , b1 }, x1 ∈ V (M ), and {a1 , a2 , b1 , b2 } ⊆ V (H) (as in
Figure 53(a)). Choose (H0 , M0 ) ∈ M to maximize M0 .
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(a) Defining (H0 , M0 )
(b) x2 satisfying (A2)
Figure 53: X1 = {x1 }, X2 = {x2 }, (Γ, X 0 ) ∈ O3 .
From planarity and (3, {a1 , b1 , a2 , b2 })-connectivity of G, we find that for any separation
(H, M ) ∈ M, M ⊆ M0 . If X 0 ⊆ V (M0 ), then (Γ, X) satisfies property (R1). Hence we
may assume that X 0 * V (M0 ), so there is some x2 ∈ X 0 with x2 ∈
/ V (M0 ), and for any
separation (H, M ) ∈ M, x2 ∈
/ V (M ).
First suppose we have a finite face with boundary D such that x2 , a2 ∈ V (D). If x2 is
cofacial with some u ∈ V (Ca ), then {u, x2 , a2 } is a 3-cut in G, leading to a 4-separation
(H, M ) ∈ M. But x2 ∈ V (M ), a contradiction. Hence we may assume that x2 is not
cofacial with any vertex of Ca . So (A1) is not satisfied for x2 .
Therefore (A2) must be satisfied, and there is some separation (Hc , Mc ) of G of order
at most 4 with {x1 , a1 , a2 , b1 , b2 } ⊆ V (Hc ), x2 ∈ V (Mc ) and |V (Hc ∩ Mc ) \ V (Ca )| ≤ 2.
Moreover, x2 ∈ V (Mc ) \ V (Hc ). Since a2 ∈ V (Hc ), we must have some vertex v ∈ V (Hc ∩
Mc ) ∩ D(x2 , a2 ]. This leads again to a separation (H, M ) ∈ M with {v, a2 } ⊆ V (H ∩ M )
113
such that x2 ∈ V (M ), a contradiction.
Hence we may assume that x2 and a2 are not incident with a common finite face.
Now (Γ, X 0 ) ∈ O3 and x2 is not cofacial with a2 , so there is some separation (Ha , Ma ) of
G order 3 such that a2 ∈ V (Ha ∩ Ma ), x2 ∈ V (Ma ) \ V (Ha ), and {a1 , a2 , b1 , b2 } ⊆ V (Ha ).
Label V (Ha ∩ Ma ) \ {a2 } as {v1 , v2 }.
If x2 is cofacial with some vertex u ∈ V (Ca ), then each of v1 , v2 must also be cofacial
with u. Now for j ∈ {1, 2} we have {u, vj , a2 } a 3-cut in G, associated with a 4-separation
(Hj , Mj ) ∈ M. But for some j ∈ {1, 2} we find x2 ∈ V (Mj ), a contradiction. Hence x2 is
not cofacial with any vertex of Ca , and (A1) is not satisfied for x2 .
So (A2) must be satisfied, and there is some separation (Hc , Mc ) of G of order at
most 4 with {x1 , a1 , b1 , a2 , b2 } ⊆ V (Hc ), x2 ∈ V (Mc ) and |V (Hc ∩ Mc ) \ V (Ca )| ≤ 2.
Moreover, x2 ∈ V (Mc ) \ V (Hc ). Choose such a separation with Mc maximal. Recalling
that x2 ∈ V (Ma ) \ V (Ha ), we find that V (Mc − Hc ) ∩ V (Ma − Ha ) 6= ∅. (See Figure
53(b).) By planarity, and maximality of Mc , we find (Hc , Mc ) ∈ M, a contradiction since
x2 ∈ V (Mc ).
(3.2)
Suppose (Γ, X 0 ) ∈ O4 or (Γ, X 0 ) satisfies (R2).
Since X2 = ∅, and we assume (Γ, X 0 ) does not satisfy (R1), we may assume there is
some x3 ∈ X 0 ∩ C(b1 , b2 ). Recalling the assumption for Case 3 that |X3 ∩ V (C)| ≤ 1, we
have X 0 ∩ C(b1 , b2 ) = {x3 }. Let Ca0 = C[x3 , a2 ] if x3 ∈ C(b1 , a2 ), and let Ca0 = C[a2 , x3 ]} if
x3 ∈ C(a2 , b2 ). Let X30 = X 0 \ {x3 }.
Since {x1 , x3 } is infeasible in Γ, there must be no path from b1 to b2 disjoint from
Ca0 ∪ C[x1 , a1 ]. So there is some 2-separation (M, N ) of G with V (M ∩ N ) = {u, v},
u ∈ C[x1 , a1 ], v ∈ Ca0 , and C[u, v] ⊆ N . (See Figure 54.) Assume the separation (M, N ) is
chosen to minimize N
If (Γ, X 0 ) ∈ O4 , we have one of the following:
(i) ∃ a finite face of G incident with some vertex in V (Ca ) and all x ∈ X30 , or
(ii) G has a separation (H, K) of order 3 with {x3 , a1 , b1 , a2 , b2 } ⊆ V (H), X30 ⊆ V (K),
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and |V (H ∩ K) \ V (Ca )| ≤ 2, or
(iii) G has a separation (H, K) of order 4 with {x3 , a1 , b1 , a2 , b2 } ⊆ V (H), X30 ⊆ V (K),
and |V (H ∩ K) \ V (Ca )| ≤ 2.
If (Γ, X 0 ) satisfies (R2), then G has a separation (H, K) of order at most 4 such that
{x3 , a1 , b1 , a2 , b2 } ⊆ V (H), X30 ⊆ V (K), and |V (H ∩ K) \ V (Ca )| ≤ 2. So we have a
separation as in (iii), with |X30 | ≥ 2. (When (Γ, X 0 ) ∈ O4 , the separation of type (iii) has
the added condition that |X30 | = 1.)
Suppose first that x3 ∈ C(b1 , a2 ), so C[a2 , x1 ] ⊆ V (M ) and C[a1 , x3 ] ⊆ V (N ) (as in Figure
54(a)). If X30 ⊆ V (M ), we have a separation (M 0 , N 0 ) of G with V (M 0 ∩ N 0 ) = {u, v, a2 , b2 }
demonstrating that (Γ, X) satisfies (R2). Similarly, if X30 ⊆ V (N ), we have a separation
(M 0 , N 0 ) of G with V (M 0 ∩ N 0 ) = {u, v, a1 , b1 }, and again (Γ, X) satisfies (R2). Hence we
may assume X30 * V (M ) and X 0 * V (N ).
If (i) above is satisfied, let D be the facial cycle of G including X30 and some vertex
ca ∈ V (Ca ). Then D * N and D * M , so by planarity we have u, v ∈ V (D), and
(Γ, X) ∈ O6 .
Therefore we assume (i) is not satisfied, so G has a separation (H, K) of order at most
4 with {x3 , a1 , b1 , a2 , b2 } ⊆ V (H), X30 ⊆ V (K), and |V (H ∩ K) \ V (Ca )| ≤ 2. Now K * M
and k * N . Choose xk ∈ X30 with xk ∈ V (N ) \ V (M ). By planarity, we must have some
vertex w ∈ V (H ∩ K) with u0 ∈ C[u, a1 ] cofacial with some vertex v 0 ∈ C[x3 , v]. This
contradicts the choice of (M, N ) to minimize N .
Now suppose that x3 ∈ C(a2 , b2 ), so C[x3 , x1 ] ⊆ V (M ) and C[a1 , a2 ] ⊆ V (N ). Let
(M 0 , N 0 ) be the separation of G with V (M 0 ∩ N 0 ) = {u, v, b2 }, C[x3 , x1 ] ⊆ V (M 0 ). If
X 0 ⊆ V (M ), then (M 0 , N 0 ) demonstrates that (Γ, X) ∈ O2 . If all vertices of X30 ∩ V (N ) lie
on the finite face including u and v, we have (Γ, X) ∈ O7 . Hence we may assume there is
some xk ∈ X30 ∩ V (N ) which is not on the finite face containing u and v.
If (i) above is satisfied, let D be the facial cycle of G including X30 and some vertex
ca ∈ V (Ca ). Choose w ∈ X30 to minimize D[w, ca ] (see Figure 54(b)). If (ii) or (iii) are
satisfied, label V (H ∩ K) = {ca , cx , w0 , w} in clockwise order along the boundary of K, with
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Figure 54: X2 = ∅, (Γ, X 0 ) ∈ O4 .
ca , cx ∈ V (C), letting ca = cx or w0 = w as appropriate if the separation is of order 3. Let
D be the facial cycle of G including w and ca , such that D[w, ca ] ⊆ H and D[ca , w] ⊆ K.
Recall that for x = x1 and each y ∈ X30 we have either a face satisfying (A1) above, or
a separation of G satisfying (A2). In either case, we find that some vertex u0 ∈ C[u, a1 ]
must be cofacial with w, or with some v 0 ∈ C[a1 , ca ]. But u0 cofacial with v 0 contradicts our
choice of (M, N ), so we have u0 cofacial with w. We thus have a 4-separation (M 00 , N 00 ) with
V (M 00 ∩ N 00 ) = {u0 , w, ca , b2 }, X ⊆ V (M 00 ), and {a1 , a2 , b1 , b2 } ⊆ N 00 , so (Γ, X) satisfies
(R1).
(3.3)
Suppose (Γ, X 0 ) ∈ O7 .
Let (H, K) be the separation and D the cycle of G satisfying the requirements of O7 .
That is, for some i ∈ {1, 2}, the following hold
1. V (H ∩ K) = {u, v, w}, w ∈ C(b3−i , ai ) ∪ C(ai , bi ), and u, v ∈ V (D)
2. {a1 , a2 , b1 , b2 } ⊆ V (H), X 0 ∩ V (K) * V (H),
3. if w ∈ C(b3−i , ai ), then u ∈ C(w, ai ] and X 0 ⊆ V (K ∪ D ∪ C(b3−i , w]), if w ∈ C(ai , bi ),
then u ∈ C[ai , w) and X 0 ⊆ V (K ∪ D ∪ C[w, bi )), and
4. if w ∈ {b3−i , bi }, then X 0 ⊆ V (K ∪ D).
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If X 0 ⊆ V (K ∪ D), then we have a cycle D 0 in G such that X 0 ⊆ Int(D 0 ) and
{a1 , b1 , a2 , b2 } ∩ Int(D 0 ) = ∅, and we apply Lemma 5.3. Hence we may assume X 0 *
V (K ∪ D), so X 0 ∩ V (C) \ V (K) 6= ∅, and w ∈
/ {b1 , b2 } (by 4 above).
Since X2 = ∅, we must have some x3 ∈ X 0 ∩ C(b1 , b2 ), with x3 ∈
/ V (K). Recalling that
|X3 ∩ V (C)| ≤ 1, we have X 0 ∩ C(b1 , b2 ) = {x3 } (Figure 55).
Let X30 = X 0 \ {x3 }. If X30 ⊆ V (K) or X30 ⊆ V (D), we find (Γ, X 0 ) ∈ O4 , and apply the
arguments of Case (3.2) above. We may assume, then, that X30 * V (K) and X30 * V (D),
so |X30 | ≥ 2. Hence |X 0 | > 3, and we may apply Lemma 5.3 to (Γ, X 0 ) to find that (Γ, X 0 )
satisfies (R2) or (Γ, X 0 ) ∈ O4 . We then apply the arguments of Case (3.2) above.
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Figure 55: Where X2 = ∅, (Γ, X 0 ) ∈ O7 .
5.3
Remaining inductive cases
This section considers the remaining cases, where X ∩ V (C) = ∅, and (Γ, X 0 ) ∈
/ O5 ∪ O6 .
We again use the definitions and lemmas associated with maximizing paths. Our first
proposition is a technical observation, which will be useful in combining results on subsets
of X.
Proposition 5.5. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C. Let X ⊆
V (G) \ {a1 , b1 , a2 , b2 }, with |X| ≥ 3, X ∩ V (C) = ∅, and X infeasible in Γ. Let P ∈ P(Γ,X)
be a maximizing path, and assume P is from a1 to x1 .
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Figure 56: Separation with X 0 * V (LP )
Let X 0 = X \ {x1 }. Suppose X 0 ∩ V (LP ) 6= ∅ =
6 X 0 \ V (LP ), and x2 ∈ X 0 \ V (LP ). If
there is a separation (H, K) of G of order at most 4 with {a1 , b1 , a2 , b2 } ⊆ V (H), (X 0 ∩
V (LP )) ∪ {x1 } ⊆ V (K), and x2 ∈
/ V (K), then either (Γ, X) ∈ O1 ∪ O2 , or (H, K) is of
order 4, and (Γ, X) satisfies property (R1).
Proof. Let Y = X \ V (LP ). Then by Lemma 3.5, G has a facial cycle D and a 3-separation
(G1 , G2 ) with V (G1 ∩ G2 ) = {u, r, w} such that
(a) u ∈ V (P ), {r, w} ⊆ V (D ∩ MP ), and D[w, r] = MP [r, w],
(b) {a1 , b1 , a2 , b2 } ⊆ V (G1 ), Y ⊆ V (G2 ), and
(c) Y ⊆ D[r, w].
If X 0 ⊆ V (D), then we have (Γ, X) ∈ O1 , so we may assume there is some x3 ∈ X 0
with x3 ∈
/ D[r, w]. We may further assume, by symmetry, that x2 ∈ D(x1 , w]. Note from
(3, {a1 , b1 , a2 , b2 })-connectivity of G that u 6= x1 , so we have x1 ∈
/ V (G1 ).
Suppose there is a separation (H, K) of G of order at most 4 with {a1 , b1 , a2 , b2 } ⊆ V (H),
(X 0 ∩ V (LP ) ∪ {x1 }) ⊆ V (K), and x2 ∈
/ V (K). Letting Ki = K ∩ Gi for i ∈ {1, 2}, we have
a separation (K1 , K2 ) of K, with x2 ∈ V (K1 ) \ V (K2 ), and x1 ∈ V (K2 ) \ V (K1 ). That is,
V (K1 ∩ K2 ) is a cutset in K. There must be some h1 , h2 ∈ V (H ∩ K) with h1 ∈ D(x1 , x2 )
and h2 ∈ V (P ).
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Since x3 ∈
/ V (D), |V (H ∩ K) ∩ V (MP )| = 2, so |V (H ∩ K)| = 4. Let V (H ∩ K) =
{h1 , h2 , h3 , h4 }. Now one of h3 , h4 must be cofacial with h1 , so we may assume h4 ∈
MP [r, w]. By (3, {a1 , b1 , a2 , b2 })-connectivity of G, we have h3 ∈ MP [b2 , r). By planarity
{h3 , h4 , u, w} is a 4-cut in G, and by Lemma 2.11 we find (Γ, X) ∈ O2 or satisfies (R1).
Lemma 5.6. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C. Let X ⊆
V (G) \ {a1 , b1 , a2 , b2 }, with |X| ≥ 3, X ∩ V (C) = ∅, and X infeasible in Γ. Let P ∈ P(Γ,X)
be a maximizing path, and assume P is from a1 to x1 ∈ X. If (Γ, X \ {x1 }) ∈ O1 , then
either (Γ, X) satisfies property (R1) or (Γ, X) ∈ O1 ∪ O2 ∪ O3 ∪ O5 ∪ O7 .
Proof. Let X 0 = X \ {x1 }, and label X 0 = {x2 , . . . , xn }. By definition of (Γ, X 0 ) ∈ O1 , all
x ∈ X 0 are incident with a common finite face, or X 0 ⊆ V (C[b1 , b2 ]), or X 0 ⊆ V (C[b2 , b1 ]).
Since X ∩V (C) = ∅, we have a finite face incident with all vertices of X 0 . Let D be the cycle
bounding this face, so X 0 ⊆ V (D). We may assume x1 ∈
/ V (D); otherwise (Γ, X) ∈ O1 .
Since X is not feasible in Γ, we may assume X 0 ∩ R(P ) = ∅. So for any x ∈ X 0 ∩ A(P ),
we have either x ∈ V (MP ) or there is some 2-cut Z ⊆ V (MP ) in LP separating x from a2 .
If D ∩P 6= ∅ and X 0 ⊆ V (MP ), then by Lemma 3.12 we have (Γ, X) ∈ O1 ∪O2 ∪O3 ∪O7 ,
or satisfies property (R1). So we assume either D ⊆ LP or X 0 * V (LP ).
Case 1
Suppose X 0 * V (LP ).
Let Y = X \ V (LP ). From Lemma 3.5, G has a facial cycle F and a separation (G1 , G2 )
such that: V (G1 ∩ G2 ) = {u, r, w} with r, w ∈ V (F ); {a1 , b1 , a2 , b2 } ⊆ V (G1 ), Y ⊆ V (K);
Y ⊆ F [r, w]. (See Figure 57.) Since x1 ∈
/ V (D), we have F 6= D. For any y ∈ Y \ {x1 },
we have y ∈ V (D ∩ F ). Since G is (3, {a1 , b1 , a2 , b2 })-connected, |Y \ {x1 }| = 1. That is,
|X 0 \ V (LP )| = 1. Let X 0 \ V (LP ) = {x2 }.
Since |X 0 | ≥ 2, we have X 0 ∩ V (LP ) 6= ∅, and hence V (D ∩ LP ) 6= ∅. Let d1 , d2 ∈
V (MP ∩ D) be chosen so that D ∩ MP = MP [d1 , d2 ]. Assume (G1 , G2 ) is chosen to minimize
P [a1 , u]. From planarity, we find u ∈ V (D). By symmetry, we may assume w ∈ V (D), so
D[w, u] is a path to the right of P , and d1 = w.
Let X̃ = X \ {x2 }. We have established D ∩ P 6= ∅, and hence X̃ \ {x1 } ⊆ V (MP ). By
Lemma 3.12 we have (Γ, X̃) ∈ O1 ∪ O2 ∪ O3 ∪ O7 , or satisfying property (R1).
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Figure 57: Example of (Γ, X 0 ) ∈ O1 with X 0 * V (LP )
If (Γ, X̃) ∈ O1 , we must have X̃ ⊆ V (F ) and |X| = 3, and hence (Γ, X) ∈ O1 . If
(Γ, X̃) ∈ O2 or satisfies (R1), we have some separation (H, K) of G of order at most 4, with
{a1 , b1 , a2 , b2 } ⊆ V (H) and X̃ ⊆ V (K). If x2 ∈ V (K), then clearly (Γ, X) ∈ O2 or satisfies
(R1). If x2 ∈
/ V (K), apply Proposition 5.5 to find (Γ, X) ∈ O1 ∪ O2 , or satisfies (R1).
Suppose (Γ, X̃) ∈ O3 . Noting that no vertex of P is shares a finite face of G with a2 ,
we find that (Γ, X̃) ∈ O3 with respect to a1 . If a1 ∈ V (D), then x2 is cofacial with a1 , and
(Γ, X) ∈ O3 . Otherwise, there must be some 3-separation (H, K) of G with V (H ∩ K) =
{a1 , h1 , h2 }, {a1 , b1 , a2 , b2 } ⊆ V (H), and x1 ∈ V (K). Since each of h1 , h2 is cofacial with
a1 , we may assume h1 ∈ MP [b2 , r] and h2 ∈ MP [d2 , b1 ]. This gives X ⊆ V (K), and
(Γ, X) ∈ O2 ∩ O3 .
Finally, suppose (Γ, X̃) ∈ O7 . Then there is a 3-separation (H, K) of G with V (H ∩K) =
{h1 , h2 , h3 } such that h1 ∈ MP [b2 , r], h2 ∈ MP [d1 , d2 ], and h3 ∈ P ∩ D, where either h1 = b2
or h1 , h3 ∈ V (C). In this case, x2 ∈ V (K), and we have (Γ, X) ∈ O7 .
Case 2
Suppose X 0 ⊆ V (LP ), and there is some xi ∈ X 0 \ V (MP )
Here D ∩ LP * MP , and we find that D is contained in a unique block of LP . Label
this block BD , and note that X 0 ⊆ V (BD ). By Lemma 3.7, if there is some cutvertex of
LP separating X 0 from a2 , then either (Γ, X) satisfies property (R1) or (Γ, X) ∈ O1 ∪ O2 ∪
O3 ∪ O5 . Hence we may assume there is no such cutvertex, and a2 ∈ V (BD ). That is,
xi ∈ A(P ).
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Since xi ∈
/ V (MP ), there is a 2-cut Z ∈ V (MP ) in LP separating xi from a2 . Choose Z
as the maximal such cut, and label Z = {z1 , z2 } with z1 left of z2 . Let Bz be the Z-bridge
of LP containing xi .
We claim that X 0 ⊆ V (Bz ). To see this, suppose not. Then V (D) * V (Bz ), so Z
separates D, and we must have Z ⊆ V (D ∩ MP ). Now if there is some xj ∈ X 0 with
xj ∈
/ V (Bz ), we have xj ∈ D[z2 , z1 ]. Since xj ∈
/ V (MP ), there is some 2-cut Z 0 ⊆ V (MP )
in LP separating xj from a2 . But now Z 0 also separates X 0 ∩ V (Bz ) from a2 , violating the
maximality of Z.
We now have xi ∈
/ V (MP ), and a 2-separation (H, K) of LP with V (H ∩ K) = Z,
X \ {x1 } ⊆ V (K), and {b1 , a2 , b2 } ⊆ H. By Lemma 3.14, either (Γ, X) satisfies property
(R1) or (Γ, X) ∈ O1 ∪ O2 ∪ O3 ∪ O5 .
Case 3
Suppose D ⊆ LP and X 0 ⊆ V (MP ).
Assume X 0 = {x2 , . . . , xn } is labeled in order from left to right along MP . If X 0 =
{x2 , x3 } and x2 x3 is a single edge, we may choose a different facial cycle D0 of G with
X 0 ⊆ V (D 0 ) and D 0 * LP , and apply the arguments of earlier cases. Hence we may assume
this is not the case, and {x2 , xn } is a 2-cut in LP . Let B be the {x2 , xn }-bridge of LP
containing MP [x2 , xn ].
If both x2 and xn are cofacial with x1 , we have a 3-cut {x1 , x2 , xn } in G showing
(Γ, X) ∈ O2 . We assume this is not the case, so there is some path from P [a1 , x1 ) to
MP (x2 , xn ). Let S be a path from s0 ∈ P [a1 , x1 ) to s ∈ MP (x2 , xn ), chosen to minimize
P [a1 , s0 ]. Assume by symmetry that S falls to the right of P , and further assume S is chosen
so that MP [s, xn ] is contained in a facial cycle also containing s0 . Since we assume X 0 is
not contained in a facial cycle, we have x2 left of s. (See Figure 58.)
If there is a separation (H, K) of G with V (H ∩K) = {s0 , x2 , xn }, {a1 , b1 , a2 , b2 } ⊆ V (H)
and X ⊆ V (K), then (Γ, X) ∈ O2 . Hence we may assume there is no such separation, so
there is some path T from P (s0 , x1 ] to P [a1 , s0 )∪MP [b2 , x2 ). By Lemma 3.2 (every (P ∪LP )bridge has an attachment in LP ), there is a path T from t0 ∈ P (s0 , x1 ] to t ∈ MP [b2 , x2 ).
Choose such a T to minimize MP [b2 , t], and subject to this minimizing P [s0 , t0 ].
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Figure 58: Example of (Γ, X 0 ) ∈ O1 with X 0 ⊆ V (MP )
Suppose t is cofacial with x2 . Then there is 4-separation (M, N ) of G with V (M ∩ N ) =
{x2 , xn , s0 , t}, X 0 ⊆ V (N ), and {a1 , b1 , a2 , b2 } ⊆ V (M ). If P [t0 , x1 ] ⊆ V (N ), then by Lemma
2.11 (4-separations in G), (Γ, X) ∈ O2 , or (Γ, X) is reducible.
So we may assume that either t is not cofacial with x2 , or for any such separation,
x1 ∈
/ V (N ). Therefore there is some path W from w0 ∈ P [s0 , x1 ] to w ∈ MP (t, x2 ). We claim
that we may choose W such that w0 6= s0 . To see this, suppose not. Then any path from
P (s0 , x1 ] to MP [t, s) has endpoint t, and {s0 , t} is a 2-cut in G violating (3, {a1 , b1 , a2 , b2 })connectivity.
By Corollary 3.13, for any xk ∈ X 0 we have (Γ, {x1 , xk }) ∈ Θ1 ∪ Θ2 ∪ Θ3 , or (Γ, {x1 , xk })
satisfies property (R1). But (Γ, {x1 , xn }) ∈
/ Θ1 , so (Γ, {x1 , xn }) ∈ Θ2 ∪ Θ3 , or satisfies (R1).
(3.1)
Suppose (Γ, {x1 , xn }) ∈ Θ2 .
Here G has a 3-separation (H, K) with {a1 , b1 , a2 , b2 } ⊆ V (H), {x1 , xn } ⊆ V (K). Let
V (H ∩ K) = {h1 , h2 , h3 }. We may assume h2 ∈ MP [xn , b1 ] and h3 ∈ P [a1 , x1 ]. If h2 6= xn ,
we must have h1 ∈ MP [b2 , xn ]. If h2 = xn , then by planarity and since h1 must be cofacial
with h2 , we find h1 ∈ V (D), and again h1 ∈ MP [b2 , xn ].
Now h1 and h3 share a finite face to the left of P , and h2 and h3 share a finite face to the
right of P . From the existence of paths S and T , we find h2 ∈ MP [b2 , t], so MP [x2 , xn ] ⊆ K.
Hence X 0 ∈ V (K) and (Γ, X) ∈ O2 .
(3.2)
Suppose (Γ, {x1 , xn }) ∈ Θ3 .
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For each j ∈ {1, n}, either xj is cofacial with a1 or G has a separation (Hj , Mj ) of order
3 such that a1 ∈ V (Hj ∩ Mj ), xj ∈ V (Mj ), {a1 , a2 , b1 , b2 } ⊆ V (Hj ). If there is a such a
separation (Hn , Mn ), we assume B * Mn ; otherwise X 0 ⊆ V (Mn ), and (Γ, X) ∈ O3 . So
there is some vertex v ∈ V (H ∩ K) ∩ V (MP (x2 , xn ]). Since v must be cofacial with a1 , this
forces s0 = a1 .
We now have either s0 = a1 , or xn , s0 , and a1 share a common finite face.
Suppose first that s0 = a1 . We may assume x2 is not cofacial with s0 , or we find (Γ, X) ∈
O3 . So there is a path R from r 0 ∈ P (a1 , x1 ] to r ∈ MP (x2 , s]. Choose R such that
each vertex of MP [r, s], is cofacial with s0 = a1 . From (3, {a1 , b1 , a2 , b2 })-connectivity of G,
Lemma 3.3, and the existence of paths T, W, and R, we see that any 3-separation (H1 , K1 )
must have MP [t, r] ⊆ V (K1 ). If this holds, then each vertex xk ∈ X 0 is either cofacial with
a1 or contained in V (K1 ), and (Γ, X) ∈ O3 . We assume, then, that there is no separation
(H1 , K1 ), and hence x1 is cofacial with a1 . Therefore, both of R and W are right of P .
By (3, {a1 , b1 , a2 , b2 })-connectivity of G, {t, t0 } is not a 2-cut in G, so we find t0 = x1 . We
may choose W with w0 6= x1 ; otherwise there is 4-separation (M, N ) of G with V (M ∩ N ) =
{x2 , xn , a1 , x1 }, X ⊆ V (N ), and {a1 , b1 , a2 , b2 } ⊆ V (M ), and by Lemma 2.11 (4-separations
in G), (Γ, X) ∈ O2 , or (Γ, X) is reducible.
Since x1 ∈
/ V (C), there is a path T0 from a1 to t0 ∈ MP [b2 , t]. We assume there is no
edge a1 b2 , so t0 6= b2 . Let Q := T0 ∪MP [t0 , t]∪T . If there are disjoint paths from b2 to w and
from a2 to x2 , each disjoint from MP [t0 , t] ∪ MP [xn , b1 ], then we have W ∪ R ⊆ V (LQ ), and
x2 ∈ R(Q). Hence X is feasible in Γ, a contradiction. So these paths do not exist, and there
must be some separation (M, N ) of LP , of order at most 3, with {t0 , b2 , a2 , b1 } ⊆ V (M )
and {t, w, x2 , xn } ⊆ V (N ). By planarity, we may assume MP [x2 , xn ] ⊆ N , so X 0 ⊆ V (N ).
We may assume there are vertices m1 , m2 ∈ V (M ∩ N ) with m1 ∈ MP [t0 , t] and m2 ∈
MP [xn , b1 ]. If |V (M ∩ N )| = 3, label the third vertex as m3 . By planarity, we have one of
the following:
(i) m1 shares a finite face of LP with m2 , or
(ii) m3 shares a face of LP with each of m1 and m2 , or
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(iii) m3 ∈ MP (t, w] shares a finite face of LP with m2 .
In case (i), there is a separation (M 0 , N 0 ) of G with V (M 0 ∩ N 0 ) = {m1 , m2 , a1 },
{a1 , b1 , a2 , b2 } ⊆ V (M 0 ), X ⊆ V (N 0 ), and hence (Γ, X) ∈ O2 ∩ O3 . In case (ii), there
is a separation (M 0 , N 0 ) of G with V (M 0 ∩ N 0 ) = {m1 , m2 , m3 , a1 }, {a1 , b1 , a2 , b2 } ⊆ V (M 0 ),
X ⊆ V (N 0 ), and hence (Γ, X) satisfies (R1). In case (iii), we have m3 cofacial with x1 , and
there is a separation (M 0 , N 0 ) of G with V (M 0 ∩ N 0 ) = {m2 , m3 , x1 , a1 }, {a1 , b1 , a2 , b2 } ⊆
V (M 0 ), X ⊆ V (N 0 ), and again (Γ, X) satisfies (R1).
Now suppose s0 6= a1 . Then xn , s0 , and a1 share a common finite face of G. Since
xn ∈
/ V (C), there is a path S0 from a1 to s0 ∈ MP (xn , b1 ]. We assume there is no edge
a1 b1 , so s0 6= b1 . Let Q := S0 ∪ MP [xn , s0 ]. If there are disjoint paths from a2 to w and
from b1 to x2 , each disjoint from MP [b2 , t] ∪ MP [xn , s0 ], then we have T ∪ S ⊆ V (LQ ).
Path W shows x1 ∈ R(Q), and hence X is feasible in Γ, a contradiction. So these paths
do not exist, and there must be some separation (M, N ) of LP , or order at most 3, with
{b2 , a2 , b1 , s0 } ⊆ V (M ) and {t, w, x2 , xn } ⊆ V (N ).
We may assume there are vertices m1 , m2 ∈ V (M ∩ N ) with m1 ∈ MP [b2 , t] and m2 ∈
MP [xn , s0 ]. If |V (M ∩ N )| = 3, label the third vertex as m3 . As before, we have one of the
following:
(i) m1 shares a finite face of LP with m2 , or
(ii) m3 shares a face of LP with each of m1 and m2 , or
(iii) m3 ∈ MP (t, w] shares a finite face of LP with m2 .
In case (i), there is a separation (M 0 , N 0 ) of G with V (M 0 ∩ N 0 ) = {m1 , m2 , a1 },
{a1 , b1 , a2 , b2 } ⊆ V (M 0 ), X ⊆ V (N 0 ), and hence (Γ, X) ∈ O2 ∩ O3 . In case (ii), there
is a separation (M 0 , N 0 ) of G with V (M 0 ∩ N 0 ) = {m1 , m2 , m3 , a1 }, {a1 , b1 , a2 , b2 } ⊆ V (M 0 ),
X ⊆ V (N 0 ), and hence (Γ, X) satisfies (R1). In case (iii), we have m3 cofacial with x1 , and
there is a separation (M 0 , N 0 ) of G with V (M 0 ∩ N 0 ) = {m2 , m3 , x1 , a1 }, {a1 , b1 , a2 , b2 } ⊆
V (M 0 ), X ⊆ V (N 0 ), and again (Γ, X) satisfies (R1).
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Figure 59: (Γ, X 0 ) ∈ O1 , X 0 ⊆ V (MP ), (Γ, {x1 , x2 }) reducible
(3.3)
Suppose (Γ, {x1 , xn }) satisfies property (R1).
Here G has some separation (H, K) of order 4 such that {a1 , b1 , a2 , b2 } ⊆ V (H),
{x1 , xn } ⊆ V (K). Choose such a separation with K maximal. Let V (H∩K) = {h1 , h2 , h3 , h4 }.
If V (MP [x2 , xn ]) ⊆ V (K), then (Γ, X) satisfies (R1). So we assume not, and that h1 ∈
MP (x2 , xn ]. Further, there is some h ∈ V (H ∩ K) with h ∈ MP [xn , b1 ]. If we may choose
h1 6= xn , then we have h2 ∈ MP [xn , b1 ]. If h1 = xn , then by planarity and since h1 must be
cofacial with h2 , we find h2 ∈ V (D), and again h2 ∈ MP [xn , b1 ].
Now by planarity and the existence of paths S and T , we have h3 ∈ P [a1 , s0 ], sharing a
face to the right of P with h2 . (See Figure 59.) Now h4 is cofacial with each of h1 and h3 .
Since x1 ∈ K and b2 ∈ H, we must have h4 ∈ MP [b2 , t] ∪ T ∪ P [t0 , x1 ]. By the existence of
path T , we see that h1 is not cofacial with any vertex of MP [b2 , t), so h4 ∈ T ∪ P [t0 , x1 ].
Now by planarity, since h4 shares a finite face of G with h1 , we find that h4 also shares a
finite face with d1 , and G has a 4-separation (H 0 , K 0 ) with V (H 0 ∩ K 0 ) = {h1 , h2 , h3 , d1 },
demonstrating that (Γ, X) satisfies (R1).
Lemma 5.7. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C. Let X ⊆
V (G) \ {a1 , b1 , a2 , b2 }, with |X| ≥ 3, X ∩ V (C) = ∅, and X infeasible in Γ. Let P ∈ P(Γ,X)
be a maximizing path, and assume P is from a1 to x1 . If (Γ, X \ {x1 }) ∈ O2 , then (Γ, X)
satisfies (R1) or (Γ, X) ∈ O1 ∪ O2 ∪ O3 ∪ O5 .
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Proof. Let X 0 = X \ {x1 }. We have (Γ, X 0 ) ∈ O2 , so G has a separation (G1 , G2 ) of order 3
such that {a1 , b1 , a2 , b2 } ⊆ V (G1 ) and X 0 ⊆ V (G2 ). We may assume the separation (G1 , G2 )
is associated with a 3-cut in G; otherwise we have all vertices of X 0 cofacial, and we apply
Lemma 5.6. Choose such a separation to maximize G2 , and let V (G1 ∩ G2 ) = {g1 , g2 , g3 }.
We assume x1 ∈
/ V (G2 ), or we have (Γ, X) ∈ O2 .
Case 1
Suppose X 0 \ V (LP ) 6= ∅.
Let Y := X \ V (LP ), and Y 0 := X 0 \ V (LP ) = Y \ {x1 }. By Lemma 3.5, G has a
facial cycle D and a separation (H, K) such that: V (H ∩ K) = {u, r, w} with r, w ∈ V (D);
{a1 , b1 , a2 , b2 } ⊆ V (H), Y ⊆ V (K); Y ⊆ D[r, w].
If X 0 ⊆ D[r, w], we have (Γ, X) ∈ O1 ∩ O2 . So we assume X 0 * D[r, w], and there is
some x2 ∈ X 0 with x2 ∈ V (LP ) \ {r, w}. So we must have |V (G1 ∩ G2 ) ∩ V (MP )| ≥ 2. Label
V (G1 ∩ G2 ) ∩ V (MP ) as {g1 , g2 }.
Since x1 ∈
/ V (G2 ), there is some separation (K1 , K2 ) of K with V (K1 ∩K2 ) ⊆ V (G1 ∩G2 ),
{u, x1 } ⊆ V (K1 ), Y 0 ⊆ V (K2 ), and x1 ∈
/ V (K2 ). Since G is (3, {a1 , b1 , a2 , b2 })-connected,
|V (K1 ∩ K2 )| ≥ 2. But this requires (G1 , G2 ) to be a separation of order 4, a contradiction.
Case 2
Suppose X 0 ⊆ V (LP ).
By Lemma 3.7, if there is some cutvertex of LP separating X 0 from a2 , then either
(Γ, X) satisfies property (R1) or (Γ, X) ∈ O1 ∪ O2 ∪ O3 ∪ O5 . Hence we may assume there
is no such cutvertex. Since G is (3, {a1 , b1 , a2 , b2 })-connected, we must have X 0 ⊆ A(P ).
Without loss of generality, assume g1 , g2 ∈ V (MP ). Suppose g1 and g2 do not share a
finite face of LP , so g3 ∈ V (LP ). In this case, MP [g1 , g2 ] is contained in a facial boundary
of G. Since G is (3, {a1 , b1 , a2 , b2 })-connected, there can be no 2-cut of LP contained in
MP [g1 , g2 ], and we must have X 0 ⊆ V (MP ); otherwise X is feasible in Γ. By Lemma 3.12,
(Γ, X) ∈ O1 ∪ O2 ∪ O3 ∪ O7 , or satisfies property (R1).
We now assume g1 shares a finite face of LP with g2 , so {g1 , g2 } is a 2-cut in LP . Let
g1 be left of g2 along MP . If there is some xk ∈
/ V (MP ), then by Lemma 3.14 either (Γ, X)
satisfies property (R1) or (Γ, X) ∈ O1 ∪ O2 ∪ O3 ∪ O5 . We assume, then, that X 0 ⊆ V (MP ).
Label X 0 = {x2 , . . . , xn }, in order from left to right along MP .
126
By (3, {a1 , b1 , a2 , b2 })-connectivity of G, {g1 , g2 } is not a 2-cut in G, so there is some path
S from s0 ∈ P [a1 , x1 ] to s ∈ MP (g1 , g2 ). Choose such a path to minimize P [a1 , s0 ]. Assume
by symmetry that S falls to the right of P , and further assume S is chosen so MP [s, g2 ]
is contained in a facial cycle also containing s0 . Since we assume X 0 is not contained in a
facial cycle, we have x2 left of s. Furthermore, since g3 is cofacial with each of g1 , g2 , we
have g3 ∈ {s0 , s}.
If there is a separation (H, K) of G with V (H ∩ K) = {s0 , g1 , g2 }, {a1 , b1 , a2 , b2 } ⊆ V (H)
and X ⊆ V (K), then (Γ, X) ∈ O2 . Hence we may assume there is no such separation, so
there is some path T from P (s0 , x1 ] to P [a1 , s0 )∪MP [b2 , g1 ). By Lemma 3.2 (every (P ∪LP )bridge has an attachment in LP ), there is a path T from t0 ∈ P (s0 , x1 ] to t ∈ MP [b2 , g1 ).
Choose such a T to minimize MP [b2 , t], and subject to this minimizing P [s0 , t0 ].
Suppose t is cofacial with g1 . By our choice of S and Lemma 3.3 (controlling paths
within (P ∪ LP )-bridges), there is some vertex v ∈ P [a1 , s0 ] cofacial with both t and g2 .
Hence there is 4-separation (M, N ) of G with V (M ∩ N ) = {g1 , g2 , v, t}, X ⊆ V (N ), and
{a1 , b1 , a2 , b2 } ⊆ V (M ). If P [t0 , x1 ] ⊆ V (N ), then by Lemma 2.11 (4-separations in G),
(Γ, X) ∈ O2 , or (Γ, X) is reducible. So we may assume that either t is not cofacial with g1 ,
or for any such separation x1 ∈
/ V (N ). Therefore there is some path W from w0 ∈ P [s0 , x1 ]
to w ∈ MP (t, g2 ). We claim that we may choose W such that w0 6= s0 . To see this, suppose
not. Then any path from P (s0 , x1 ] to MP [t, s) has endpoint t, and {s0 , t} is a 2-cut in G
violating (3, {a1 , b1 , a2 , b2 })-connectivity.
If x1 shares a finite face of G with each of g3 and g1 , then we have a 4-separation (G01 , G02 )
of G with V (G01 ∩ G02 ) = {g1 , g2 , g3 , x1 }, and by Lemma 2.11, (Γ, X) ∈ O2 or satisfies (R1).
We assume this is not the case, so either (i) W is right of P , or (ii) g3 = s, and there is a
path R to the left of P , from r 0 ∈ P [w0 , x1 ) to s. Consider case (ii). Suppose W is chosen to
minimize P [w0 , r 0 ]. G is (3, {a1 , b1 , a2 , b2 })-connected, so for any vertex v ∈ P [w0 , r 0 ], the set
{v, s} is not a 2-cut in G. Hence there must be a path P 0 from p0 ∈ P [a1 , w0 ) to p ∈ P (r 0 , x1 ],
contained in some (P ∪ LP )-bridge and disjoint from P ∪ LP . But together with paths W
and R, this contradicts Lemma 3.3. Hence case (ii) cannot occur, and we have W right of
P . Recalling that g3 ∈ {s0 , s} is cofacial with g1 , we have w ∈ MP (t, g1 ] ∪ {s}.
127
By Corollary 3.13, for any xk ∈ X 0 we have (Γ, {x1 , xk }) ∈ Θ1 ∪ Θ2 ∪ Θ3 , or (Γ, {x1 , xk })
satisfies property (R1). Note from path W that x1 is not cofacial with xn . Therefore
(Γ, {x1 , xn }) ∈
/ Θ1 , so (Γ, {x1 , xn }) ∈ Θ2 ∪ Θ3 , or satisfies (R1).
(2.1)
Suppose (Γ, {x1 , xn }) ∈ Θ2 .
Here G has a 3-separation (H, K) with {a1 , b1 , a2 , b2 } ⊆ V (H), {x1 , xn } ⊆ V (K). If
X 0 ⊆ V (K), then (Γ, X) ∈ O2 , so we assume X 0 * V (K).
Let V (H ∩ K) = {h1 , h2 , h3 }. Since X 0 * V (K), we may assume h1 ∈ MP (x2 , xn ]. We
further assume h3 ∈ P [a1 , x1 ], and note V (H ∩ K) ∩ V (MP [xn , b1 ]) 6= ∅. By the existence
of path S, we may assume V (H ∩ K) ∩ V (MP ) 6= {xn }, so h2 ∈ MP [xn , b1 ]. Since h2 is
cofacial with h3 , we have h3 ∈ P [a1 , s0 ]. But now we have a path from x1 to b2 along T ,
disjoint from V (H ∩ K), a contradiction.
(2.2)
Suppose (Γ, {x1 , xn }) satisfies property (R1).
Here G has some separation (H, K) of order 4 such that {a1 , b1 , a2 , b2 } ⊆ V (H),
{x1 , xn } ⊆ V (K). Choose such a separation with K maximal. By Lemma 2.8, V (H ∩
K) ∩ {a1 , b1 , a2 , b2 } 6= ∅ or |V (H ∩ K) ∩ V (C)| ≥ 2. If V (MP [x2 , xn ]) ⊆ V (K), then
(Γ, X) satisfies (R1). So we assume not, and there must be a vertex h1 ∈ V (H ∩ K) with
h1 ∈ MP (x2 , xn ].
Let V (H ∩K) = {h1 , h2 , h3 , h4 }. If X 0 ⊆ V (K), then (Γ, X) satisfies (R1). So we assume
not, and that h1 ∈ MP (x2 , xn ]. We further assume h3 ∈ P [a1 , x1 ], and note V (H ∩ K) ∩
V (MP [xn , b1 ]) 6= ∅. By the existence of path S, we may assume V (H ∩ K)∩ V (MP ) 6= {xn },
so h2 ∈ MP [xn , b1 ].
Now by planarity and the existence of paths S and T , we have h3 ∈ P [a1 , s0 ], sharing
a face to the right of P with h2 . Now h4 is cofacial with each of h1 and h3 . Since x1 ∈ K
and b2 ∈ H, we must have h4 ∈ MP [b2 , t] ∪ T ∪ P [t0 , x1 ]. By the existence of path T , we see
that h1 is not cofacial with any vertex of MP [b2 , t), so h4 ∈ T ∪ P [t0 , x1 ]. Now by planarity,
since h4 shares a finite face of G with h1 , we find that h4 also shares a finite face with g1 ,
and G has a 4-separation (H 0 , K 0 ) with V (H 0 ∩ K 0 ) = {h1 , h2 , h3 , g1 }, demonstrating that
(Γ, X) satisfies (R1).
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a1
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g1 x2
11
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Figure 60: (Γ, X 0 ) ∈ O2
(2.3)
Suppose (Γ, {x1 , xn }) ∈ Θ3 .
For each j ∈ {1, n}, either xj is cofacial with a1 or G has a separation (Hj , Kj ) of order
3 such that a1 ∈ V (Hj ∩ Kj ), xj ∈ V (Kj ), {a1 , a2 , b1 , b2 } ⊆ V (Hj ). If there is a such a
separation (Hn , Kn ), we assume X 0 * V (Kn ); otherwise X 0 ⊆ V (Kn ), and (Γ, X) ∈ O3 . So
there is some vertex v ∈ V (H ∩ K) ∩ V (MP (x2 , xn ]). Since v must be cofacial with a1 , this
forces s0 = a1 . We now have either s0 = a1 , or xn , s0 , and a1 share a common finite face.
Suppose first that s0 = a1 (Figure 60). We may assume g1 is not cofacial with s0 , or we
find (Γ, X) ∈ O3 . But each of g1 , g2 is cofacial with g3 , so we must have g3 = s, and hence
w ∈ MP (t, g1 ). Furthermore, there is a path R from r 0 ∈ P (a1 , w] to s. Note that g3 = s
together with (3, {a1 , b1 , a2 , b2 })-connectivity of G shows that G2 ⊆ LP , and G2 ∩ LP is
2-connected.
If t0 6= x1 , then the 3-separation (H1 , K1 ) exists and must have vertices h1 , h2 ∈ V (H1 ∩
K1 ), with h1 ∈ MP [b2 , t] and h2 ∈ MP [s, b1 ] sharing a finite face. But since G2 ∩ LP is
2-connected, this forces h2 ∈ MP [g2 , b1 ]. Hence X 0 ⊆ V (K1 ), and (Γ, X) ∈ O2 ∩ O3 . Hence
we may assume t0 = x1 .
Since x1 ∈
/ V (C), there is a path T0 from a1 to t0 ∈ MP [b2 , t]. We assume there is no
edge a1 b2 , so t0 6= b2 . Let Q := T0 ∪MP [t0 , t]∪T . If there are disjoint paths from b2 to w and
from a2 to x2 , each disjoint from MP [t0 , t] ∪ MP [g2 , b1 ], then we have W ∪ R ⊆ V (LQ ), and
x2 ∈ R(Q). Hence X is feasible in Γ, a contradiction. So these paths do not exist, and there
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must be some separation (M, N ) of LP , or order at most 3, with {t0 , b2 , a2 , b1 } ⊆ V (M )
and {t, w, x2 , xn } ⊆ V (N ).
We may assume there are vertices m1 , m2 ∈ V (M ∩ N ) with m1 ∈ MP [t0 , t] and m2 ∈
MP [xn , b1 ]. If |V (M ∩ N )| = 3, label the third vertex as m3 . By planarity, we have one of
the following:
(i) m1 shares a finite face of LP with m2 , or
(ii) m3 shares a face of LP with each of m1 and m2 , or
(iii) m3 ∈ MP (t, w] shares a face of LP with m2 .
In case (i), there is a separation (M 0 , N 0 ) of G with V (M 0 ∩ N 0 ) = {m1 , m2 , a1 }, contradicting our choice of (G1 , G2 ) to maximize G2 . In case (ii), there is a separation (M 0 , N 0 )
of G with V (M 0 ∩ N 0 ) = {m1 , m2 , m3 , a1 }, {a1 , b1 , a2 , b2 } ⊆ V (M 0 ), X ⊆ V (N 0 ), and hence
(Γ, X) satisfies (R1). In case (iii), we have m3 cofacial with x1 , and there is a separation
(M 0 , N 0 ) of G with V (M 0 ∩ N 0 ) = {m2 , m3 , x1 , a1 }, {a1 , b1 , a2 , b2 } ⊆ V (M 0 ), X ⊆ V (N 0 ),
and again (Γ, X) satisfies (R1).
Now suppose s0 6= a1 , so xn , s0 , and a1 share a common finite face. Since xn ∈
/ V (C),
there is a path S0 from a1 to s0 ∈ MP (g2 , b1 ]. We assume there is no edge a1 b1 , so s0 6= b1 .
Let Q := S0 ∪ MP [xn , s0 ]. If there are disjoint paths from a2 to w and from b1 to g1 ,
each disjoint from MP [b2 , t] ∪ MP [xn , s0 ], then we have T ∪ S ⊆ V (LQ ). Path W shows
x1 ∈ R(Q), Hence X is feasible in Γ, a contradiction. So these paths do not exist, and there
must be some separation (M, N ) of LP , or order at most 3, with {b2 , a2 , b1 , s0 } ⊆ V (M )
and {t, w, x2 , xn } ⊆ V (N ).
We may assume there are vertices m1 , m2 ∈ V (M ∩ N ) with m1 ∈ MP [b2 , t] and m2 ∈
MP [xn , s0 ]. If |V (M ∩ N )| = 3, label the third vertex as m3 . As before, we have one of the
following:
(i) m1 shares a finite face of LP with m2 , or
(ii) m3 shares a face of LP with each of m1 and m2 , or
130
(iii) m3 ∈ MP (t, w] shares a finite face of LP with m2 .
In case (i), there is a separation (M 0 , N 0 ) of G with V (M 0 ∩ N 0 ) = {m1 , m2 , a1 }, contradicting our choice of (G1 , G2 ) to maximize G2 . In case (ii), there is a separation (M 0 , N 0 )
of G with V (M 0 ∩ N 0 ) = {m1 , m2 , m3 , a1 }, {a1 , b1 , a2 , b2 } ⊆ V (M 0 ), X ⊆ V (N 0 ), and hence
(Γ, X) satisfies (R1). In case (iii), we have m3 cofacial with x1 , and there is a separation
(M 0 , N 0 ) of G with V (M 0 ∩ N 0 ) = {m2 , m3 , x1 , a1 }, {a1 , b1 , a2 , b2 } ⊆ V (M 0 ), X ⊆ V (N 0 ),
and again (Γ, X) satisfies (R1).
Lemma 5.8. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C. Let X ⊆ V (G) \
{a1 , b1 , a2 , b2 }, with |X| ≥ 3, X ∩ V (C) = ∅, and X infeasible in Γ. If (Γ, X \ {x1 }) ∈ O3 ,
then (Γ, X) satisfies (R1) or (Γ, X) ∈ O3 .
Proof. Let X 0 = X \ {x1 } = {x2 , . . . , xn }. We assume by symmetry that (Γ, X 0 ) ∈ O3 with
respect to a2 , so for each x ∈ X 0 , one of the following holds
(a) there is a finite face of G incident with both a2 and x, or
(b) G has a separation (H, K) of order 3 such that a2 ∈ V (H∩K), x ∈ V (K), {a1 , a2 , b1 , b2 } ⊆
V (H), or
(c) x ∈ C[b1 , b2 ].
Since X ∩ V (C) = ∅, one of (a) or (b) must hold.
By Lemma 3.1, either (Γ, X) ∈ O3 with respect to a1 , or G − V (C[b1 , b2 ]) contains a
path P from a1 to X for which a2 is not a cutvertex of LP . We may assume (Γ, X) ∈
/ O3 ,
so we have a path P ∈ P(Γ,X) from a1 to X.
If x ∈ X 0 is cofacial with a2 , then for any path P 0 from a1 to x, we find that a2 is
a cutvertex of LP 0 . Hence P 0 ∈
/ P(Γ,X) . If x ∈ X 0 is not cofacial with a2 , then G has a
separation (H, K) with V (H ∩ K) = {h1 , h2 , a2 }, x ∈ V (K), and {a1 , a2 , b1 , b2 } ⊆ V (H).
Any path P 0 from a1 to x must have V (P 0 ) ∩ {h1 , h2 , a2 } 6= ∅, and again P 0 ∈
/ P(Γ,X) . So
we may assume each path in P(Γ,X) is from a1 to x1 . Let P ∈ P(Γ,X) be a maximizing path.
Claim 1. X 0 ⊆ V (LP ).
131
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Figure 61: Where (Γ, X 0 ) ∈ O3
Proof of Claim 1 To see this, suppose on the contrary that there is some xk ∈ X 0 \V (LP ).
Let Y = X \V (LP ). Then by Lemma 3.5, G has a facial cycle D and a 3-separation (G1 , G2 )
such that: V (G1 ∩ G2 ) = {u, r, w} with r, w ∈ V (D); {a1 , b1 , a2 , b2 } ⊆ V (G1 ), Y ⊆ V (K);
Y ⊆ D[r, w]. Since a2 is not a cutvertex of LP , it is not cofacial with xk . So G has a
separation (H, K) meeting condition (b).
There are three internally disjoint paths from xk to {b1 , b2 , a1 }; D[xk , w] ∪ MP [w, b1 ],
MP [b2 , r] ∪ D[r, xk ], and a path including P [a1 , u], respectively. Hence we must have some
h1 , h2 ∈ V (H ∩ K) with h1 ∈ MP [b2 , r] and h2 ∈ MP [r, b1 ], so V (H ∩ K) = {a1 , h1 , h2 }.
But this leaves a path from xk to a1 disjoint from V (H ∩ K), a contradiction. This proves
Claim 1.
Now suppose there is some xk ∈
/ V (MP ). Any separation (H, K) meeting condition (b)
must have some V (H ∩ K) = {h1 , h2 , a2 } with h1 ∈ MP [b2 , zk ] and h2 ∈ MP [zk0 , b1 ]. But
then Zk is a 2-cut in G, which contradicts the (3, {a1 , b1 , a2 , b2 })-connectivity of G. So if
xk ∈
/ V (MP ), (b) cannot be satisfied, hence xk is cofacial with a2 .
Note that each zk , zk0 is cofacial with a2 . Let zl be the leftmost vertex of {z2 , . . . , zn } and
zr the rightmost vertex of {z20 , . . . , zn0 }. Since each of zl , zr is cofacial with a2 , {zl , zr , a2 } is
a 3-cut in LP . (See Figure 61.)
We claim that we may assume there is some path from P to MP (zl , zr ). To see this,
suppose first that X 0 * V (MP ). Then there is some Zk which is a 2-cut in LP . Since G
132
is (3, {a1 , b1 , a2 , b2 })-connected, Zk is not a 2-cut in G, and the desired path must exist.
Now suppose X 0 ⊆ V (MP ). Then X 0 ⊆ V (MP [zl , zr ]). If MP [zl , zr ] is contained in a face
boundary of G, then by Lemma 3.12, (Γ, X) ∈ O3 or (Γ, X) satisfies property (R1). So we
may assume MP [zl , zr ] is not contained in a face boundary, and again the desired path must
exist. Choose a path S from s0 ∈ V (P ) to s ∈ MP (zl , zr ) to minimize P [a1 , s0 ]. Assume
by symmetry that S falls to the right of P , and is chosen to minimize MP [s, zr ], so zr is
cofacial with s0 .
We may assume no vertex v ∈ V (P ) is cofacial with both zl and zr ; otherwise there is a
4-separation demonstrating that (Γ, X) satisfies (R1). Applying this to x1 , we find s0 6= x1 .
Applying it to s0 , we find there is some path U from u0 ∈ P (s0 , x1 ] to u ∈ MP [b2 , zl ). Choose
U to minimize MP [b2 , u]. If some vertex in MP (b2 , u) shares a finite face with a2 , let zu be
the rightmost such vertex. Otherwise, let zu = b2 .
Now assume no v ∈ V (P ) is cofacial with both zu and zr , or we again find (Γ, X) satisfies
(R1). Choose paths T from t0 ∈ P [a1 , s0 ) to t ∈ MP (zu , u] and W from w0 ∈ P (t0 , s0 ] to w ∈
MP (zr , b1 ). Let Q := T ∪ MP [t, u] ∪ U ∪ P [u0 , x1 ] (Figure 61). By our choice of T , there is a
path from zu to zl in LP , disjoint from MP [t, u]∪{a2 }, so MP [zl , s]∪S ∪W ∪MP [w, b1 ] ⊆ LQ .
By our choice of S, there is a path from a2 to xr along the facial cycle they share, disjoint
from MP [zl , s]∪S ∪W ∪MP [w, b1 ], so xr ∈ R(Q), and X is feasible in Γ, a contradiction.
We now return to using Theorem 1 inductively to complete the final cases. The proof of
Lemma 5.9 will not use maximizing paths directly. The choice of a maximizing path in the
conditions is included in order to apply previous lemmas.
Lemma 5.9. Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph with outer cycle C, and let X ⊆
V (G)\{a1 , b1 , a2 , b2 } with |X| = n ≥ 3, X∩V (C) = ∅, and X infeasible in Γ. Let P ∈ P(Γ,X)
be a maximizing path, and assume P is from {a1 , a2 } to x1 ∈ X. If (Γ, X \ {x1 }) ∈ O7 , and
Theorem 1 holds for vertex sets of size n − 1, then either (Γ, X) satisfies property (R1) or
(Γ, X) ∈ O1 ∪ O2 ∪ O3 ∪ O5 ∪ O7 .
Proof. Let X 0 = X \ {x1 }, so (Γ, X 0 ) ∈ O7 , as given in Definition 2.9. We assume by
symmetry, and since X ∩ V (C) = ∅, that G has a 3-separation (G1 , G2 ) and a cycle D
133
bounding a finite face of G such that the following hold
1. V (G1 ∩ G2 ) = {u, v, w}, w ∈ C[b2 , a1 ), and u, v ∈ V (D)
2. {a1 , a2 , b1 , b2 } ⊆ V (G1 ), X ∩ V (G2 ) * V (G1 ),
3. either w ∈ C(b2 , a1 ) and u ∈ C(w, a1 ], or w = b2 ,
4. X ⊆ V (G2 ∪ D).
Let G0 = G2 ∪ D. Let E be the facial cycle of G with v, w ∈ V (E), so that E[v, w] ⊆ G1
and E[w, v] ⊆ G2 .
Note that the subgraph G0 may have at most one cutvertex, w0 ∈ V (G2 ), with ww0 an
edge in G2 . To see this, suppose w0 is a cutvertex of G0 . Since D is 2-connected, we must
have w0 separating w from {u, v} in G0 . But since G is (3, {a1 , b1 , a2 , b2 })-connected, {w, w0 }
is not a 2-cut in G, hence ww0 must be a single edge. By a similar argument, if there is
a cutvertex w0 of G2 , there must be an edge uw0 or vw0 in G2 . If there is any separation
(H, K) of G0 of order at most 2 with x3 ∈ V (K) and x2 ∈
/ V (K), we must have either
x2 ∈ V (H ∩ K) or |V (H ∩ K) ∩ V (D)| = 2.
We assume x1 ∈
/ V (G0 ); otherwise (Γ, X) ∈ O7 . We also assume there is some x2 ∈
V (G2 ) with x2 ∈
/ V (D); otherwise (Γ, X 0 ) ∈ O1 , and by Lemma 5.6, (Γ, X) satisfies property
(R1) or (Γ, X) ∈ O1 ∪ O2 ∪ O3 ∪ O5 ∪ O7 . Since X 0 ∩ V (C) = ∅, we have x2 6= w, so {u, v, w}
is a 3-cut in G.
We assume some x3 ∈ V (D) \ V (G2 ); otherwise (Γ, X 0 ) ∈ O2 , and by Lemma 5.7, (Γ, X)
satisfies (R1) or (Γ, X) ∈ O1 ∪ O2 ∪ O3 ∪ O5 . We further assume ai ∈
/ V (D) for i ∈ {1, 2};
otherwise {ai , v, w} is a 3-cut in G, and we find (Γ, X 0 ) ∈ O3 , and by Lemma 5.8, (Γ, X)
satisfies (R1) or (Γ, X) ∈ O3 .
Let X2 = X \ {x2 } and X3 = X \ {x3 }. Since X is infeasible in Γ, we must have Xj
infeasible in Γ for each j ∈ {2, 3}. We assume Theorem 1 holds for vertex sets of size n − 1,
S
so (Γ, Xj ) ∈ 7i=1 Oi , or (Γ, Xj ) is reducible. Since X ∩V (C) = ∅, (Γ, Xj ) ∈
/ O4 , and we may
assume (Γ, Xj ) does not satisfy (R2). If (Γ, Xj ) ∈ O5 , then by Lemma 5.1, (Γ, X) satisfies
property (R1) or (Γ, X) ∈ O2 ∪ O4 ∪ O5 . If (Γ, Xj ) ∈ O6 , then by Lemma 5.2, either (Γ, X)
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(a) (Γ, X3 ) ∈ O1
(b) (Γ, X3 ) ∈ O2
Figure 62: Examples of (Γ, X 0 ) ∈ O7
is reducible or (Γ, X) ∈ O6 . We are left with the cases that (Γ, Xj ) ∈ O1 ∪ O2 ∪ O3 ∪ O7 ,
or satisfies (R1).
Case 1
Suppose (Γ, X3 ) ∈ O1 .
Each vertex of X3 appears on a common facial cycle of G. In particular, x1 is cofacial
with x2 . Since x2 ∈ V (G2 ) \ V (D), x2 6= w, and by symmetry if w = b2 , we may assume
the common facial cycle is E (Figure 62(a)). Note that X ∩ (V (D) \ {v}) = {x3 }.
Suppose we have 4 disjoint paths in G1 : P1 from a1 to x3 ; O from b1 to v; P2 from a2 to
x1 ; and C[b2 , w]. Then C[b2 , w] ∪ E[w, v] ∪ O is a path from b1 to b2 , disjoint from P1 ∪ P2 ,
and we see X is feasible in Γ. Hence no such paths exist, and we must have some separation
(H, K) of G of order at most 3 with {a1 , b1 , a2 , b2 } ⊆ V (H), {x3 , v, x1 , w} ⊆ V (K). By
planarity and (3, {a1 , b1 , a2 , b2 })-connectivity of G, we find X ⊆ V (K), and (Γ, X) ∈ O2 .
Case 2
Suppose (Γ, X3 ) ∈ O2 .
Here there is some 3-separation (H, K) of G with {a1 , b1 , a2 , b2 } ⊆ V (H) and X3 ⊆
V (K). We may assume x3 ∈
/ V (K); otherwise (Γ, X) ∈ O2 . Hence we must have V (H ∩
K) = {u0 , v 0 , w0 } with u0 ∈ D[u, x3 ], v 0 ∈ D[x3 , v], and by planarity, w0 ∈ C[b2 , w] (Figure
62(b)). So (Γ, X) ∈ O7 .
135
Case 3
Suppose (Γ, X3 ) ∈ O3 .
For some i ∈ {1, 2}, either x2 is cofacial with ai or there is a 3-separation (H2 , K2 ) of
G with ai ∈ V (H2 ∩ K2 ), {a1 , b1 , a2 , b2 } ⊆ V (H2 ) and x2 ⊆ V (K2 ). We may assume x3
is not cofacial with ai , and if there is a separation (H2 , K2 ), then x3 ∈
/ V (K2 ); otherwise
(Γ, X) ∈ O3 .
We may assume x2 is cofacial with ai ; otherwise, the separation (H2 , K2 ) exists, and
by planarity, ai ∈ V (D), a contradiction. Since x2 ∈
/ V (D) and x2 6= w, we may assume
ai , x2 ∈ V (E). Since x2 ∈
/ V (C), we assume E bounds a finite face. If a1 ∈ V (E), we
have a 2-cut {a1 , w} in G, violating (3, {a1 , b1 , a2 , b2 })-connectivity. Hence we may assume
a2 ∈ V (E) (Figure 63(a)).
Now {a2 , w} is a 2-cut in G, and there is a 4-separation (M, N ) of G with V (M ∩ N ) =
{w, a1 , b1 , a2 }, {a1 , b1 , a2 , b2 } ⊆ V (M ), and X 0 ⊆ V (N ). If x1 ∈ V (N ), then (Γ, X) satisfies
(R1). So we assume x1 ∈
/ V (N ).
Now x1 ∈
/ V (C), and G is (3, {a1 , b1 , a2 , b2 })-connected, so there is a path from a2 to x1
in N which is disjoint from C[b2 , w]∪E[w, v]. Suppose there are disjoint paths from a1 to x3 ,
and from b1 to v in M − a2 . These paths, together with those identified in N , would show X
feasible in Γ, a contradiction. Hence no such paths may exist, and we must have a separation
(M1 , M2 ) of M , of order at most 2, with {a1 , b1 , a2 } ⊆ V (M1 ) and {x3 , v, a2 } ⊆ V (M2 ).
But then we have a separation (M 0 , N 0 ) of G with V (M 0 , N 0 ) = V (M1 ∩ M2 ) ∪ {b2 }, with
{a1 , b1 , a2 , b2 } ⊆ V (M 0 ) and X ⊆ V (N 0 ), so (Γ, X) satisfies (R1).
Case 4
Suppose (Γ, X3 ) ∈ O7 .
There is some 3-separation (H, K) of G and finite facial cycle D 0 with V (H ∩ K) =
{u0 , v 0 , w0 }, {a1 , a2 , b1 , b2 } ⊆ V (H), X ∩ V (K) * V (H), satisfying Definition 2.9. We may
assume x3 ∈
/ V (K ∪ D 0 ), or we have (Γ, X) ∈ O7 . We further assume X3 * V (D 0 ) and
X3 * V (K); otherwise we have one of (Γ, X3 ) ∈ O1 or (Γ, X3 ) ∈ O2 , and apply the
arguments of those cases above.
If x2 ∈ V (K), then since x3 ∈
/ V (K), we must have |{u0 , v 0 , w0 } ∩ V (D)| ≥ 2. By
planarity, this forces u0 , v 0 ∈ V (D), so D 0 = D, and (Γ, X) ∈ O7 .
136
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a2
b2
(b) (Γ, X3 ) ∈ O7
Figure 63: Examples of (Γ, X 0 ) ∈ O7
So we may assume x2 ∈
/ V (K), and x2 ∈ V (D 0 ). Hence u0 , v 0 ∈ V (E) (Figure 63(b)).
This gives w 6= b2 , w0 ∈ C[b2 , w], and u0 = w. If s0 shares a finite face of G with v, we have
a 3-cut {w0 , v, u} and cycle D demonstrating (Γ, X) ∈ O7 . We assume this is not the case,
so v 0 ∈ E(v, w).
Let N := G2 ∪ K, so there is a separation (M, N ) of G with V (M, N ) = {u, v, v 0 , w0 }.
Now x1 ∈
/ V (C), and G is (3, {a1 , b1 , a2 , b2 })-connected, so there is a path from v 0 to x1
in K which is disjoint from C[w0 , w]. Hence there is a path from v 0 to x1 in N , disjoint
from C[w0 , w] ∪ E[w, v]. Suppose there are disjoint paths from a1 to x3 , from b1 to v,
from a2 to v 0 , and from b2 to w0 in M . These paths, together with those identified in
N , would show X feasible in Γ, a contradiction. Hence no such paths may exist, and we
must have a separation (M1 , M2 ) of M , of order at most 3, with {a1 , b1 , a2 , b2 } ⊆ V (M1 )
and {x3 , v, v 0 , w0 } ⊆ V (M2 ). But then V (M1 ∩ M2 ) is a 3-cut in G demonstrating that
(Γ, X) ∈ O2 .
Case 5
Suppose (Γ, X3 ) satisfies (R1).
There is some 4-separation (H, K) of G with {a1 , b1 , a2 , b2 } ⊆ V (H), X3 ⊆ V (K).
We assume the separation is chosen with K maximal, so by Lemma 2.8, V (H ∩ K) ∩
{a1 , b1 , a2 , b2 } 6= ∅ or |V (H ∩ K) ∩ V (C)| ≥ 2. Label V (H ∩ K) = {h1 , h2 , h3 , h4 }.
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(b)
b1
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b2
(c)
Figure 64: (Γ, X 0 ) ∈ O7 with (Γ, X3 ) reducible
We assume x3 ∈
/ V (K); otherwise (Γ, X) satisfies (R1). So we must have |V (H ∩ K) ∩
V (D)| ≥ 2.
(5.1)
Suppose w = b2 .
We assume by symmetry that h1 = b2 , h2 ∈ D[u, x3 ), h3 ∈ D(x3 , v], and h4 ∈ C[a2 , b2 ]∪
E[v, b2 ] (Figure 64(a)). We assume the 3-cut {b2 , h2 , v} does not separate x1 from x3 ;
otherwise (Γ, X) ∈ O7 . Similarly assume h4 6= b2 .
We now have disjoint paths from b2 to h3 and from h4 to x1 in K. Suppose there are
disjoint paths from a2 to h4 , from b1 to h3 , and from a1 to x3 in H − b2 . Together with the
paths in K, these would show X feasible in Γ, a contradiction. Hence no such paths exist,
138
and there must be a separation (H1 , H2 ) of H, of order at most 3, with {a1 , b1 , a2 , b2 } ⊆
V (H1 ) and {x3 , h3 , h4 , b2 } ⊆ V (H2 ). But then V (H1 ∩ H2 ) is a 3-cut in G demonstrating
that (Γ, X) ∈ O2 .
(5.2)
Suppose w 6= b2 .
Here we assume that h1 = u, h2 ∈ D(x3 , v], and h4 ∈ C[b2 , w] (Figures 64(b) and 64(c)).
Now there is a path from h3 to x1 in K, disjoint from C[h4 , w] ∪ E[w, v] ∪ D[h2 , v]. Suppose
there are disjoint paths in H from b2 to h4 , from a2 to h3 , from b1 to h2 , and from a1 to
x3 . Together with the paths in K, these show X feasible in Γ, a contradiction. Hence no
such paths exist, and there must be a separation (H1 , H2 ) of H, of order at most 3, with
{a1 , b1 , a2 , b2 } ⊆ V (H1 ) and {x3 , h2 , h3 , h4 } ⊆ V (H2 ). But then V (H1 ∩ H2 ) is a 3-cut in G
demonstrating that (Γ, X) ∈ O2 .
5.4
Proof of Theorem 1
Let Γ = (G, a1 , b1 , a2 , b2 ) be a disk graph, and let X ⊆ V (G) \ {a1 , b1 , a2 , b2 } with |X| ≥ 2.
We wish to prove that one of the following holds:
(C1) X is feasible in Γ,
(C2) (Γ, X) is reducible, or
(C3) (Γ, X) ∈
S7
i=1 Oi .
We assume |X| > 2; otherwise apply Theorem 2. We suppose that X is infeasible in Γ,
and show that (Γ, X) satisfies one of (C2) or (C3). This is done inductively, assuming that
for any Y ⊆ X with Y 6= X, (Γ, Y ) satisfies one of (C2) or (C3).
Suppose first that X ∩ V (C) 6= ∅. By Lemma 5.4, either (Γ, X) is reducible, or (Γ, X) ∈
S7
i=1 Oi .
So we may assume X ∩ V (C) = ∅.
Recall that P is defined as the collection of all paths P from ai ∈ {a1 , a2 } to xj ∈ {x1 , x2 }
in G − (C[bi , b3−i ] ∪ {x3−j }) such that a3−i is not a cutvertex of LP . From Lemma 3.1,
either P 6= ∅ or (Γ, X) ∈ O3 , so we may assume that there is a maximizing path P ∈ P,
139
from a1 to x1 ∈ X. Define X 0 = X \ {x1 }, so (Γ, X 0 ) is reducible, or (Γ, X) ∈
S7
i=1 Oi .
Since X 0 ∩ V (C) = ∅, (Γ, X 0 ) ∈
/ O4 and (Γ, X 0 ) does not satisfy property (R2).
If (Γ, X 0 ) ∈ O1 , then by Lemma 5.7 either (Γ, X) satisfies (R1) or (Γ, X) ∈ O1 ∪O2 ∪O3 ∪
S
O5 . If (Γ, X 0 ) ∈ O2 , Lemma 5.7 gives (Γ, X) reducible or (Γ, X) ∈ 7i=1 Oi . If (Γ, X 0 ) ∈ O3 ,
Lemma 5.8 gives (Γ, X) reducible or (Γ, X) ∈ O3 . If (Γ, X 0 ) ∈ O5 , we see from Lemma 5.1
that either (Γ, X) satisfies property (R1) or (Γ, X) ∈ O2 ∪ O4 ∪ O5 . If (Γ, X 0 ) ∈ O6 , Lemma
5.2 gives (Γ, X) reducible or (Γ, X) ∈ O6 .
We may now assume (Γ, X 0 ) satisfies (R1), and (Γ, X 0 ) ∈
/
S7
i=1 Oi .
From the definition
of (R1), G has some separation (H, K) of order 4 such that
(i) {a1 , b1 , a2 , b2 } ⊆ V (H) and X ⊆ V (K),
(ii) the vertices in V (H ∩ K) can be labeled as a01 , b01 , a02 , b02 such that
(a) a0i is cofacial with b0j for each i, j ∈ {1, 2},
(b) any four disjoint paths from {a1 , b1 , a2 , b2 } to {a01 , b01 , a02 , b02 } join ai to a0i and bi
to b0i respectively (i = 1, 2)
Choose such a separation with K maximal. We may assume x1 ∈
/ V (K); otherwise
(Γ, X) satisfies (R1). We may further assume there is no 3-separation (G1 , G2 ) of G with
{a1 , b1 , a2 , b3 } ⊆ V (G1 ) and K ⊆ V (G2 ); otherwise (Γ, X 0 ) ∈ O2 .
Case 1
Suppose {a1 , a2 } ∩ V (H ∩ K) 6= ∅.
Assume by symmetry that a1 = a01 .
Let E1 be the facial cycle of G shared by b01 and a02 , and let E2 be the facial cycle
of G shared by b02 and a02 . Note by the maximality of K that each of E1 , E2 bounds a
finite face (Figure 65(b)). By (3, {a1 , b1 , a2 , b2 })-connectivity of G and our assumption that
(Γ, X 0 ) ∈
/ O3 , we have a1 ∈
/ V (Ei ) for i ∈ {1, 2}. By (3, {a1 , b1 , a2 , b2 })-connectivity of G,
b02 ∈
/ V (E1 ) and b01 ∈
/ V (E2 ).
(1.1)
Suppose one of the following holds (Figure 65(a)):
(i) there is a finite face of G incident with both a1 and x1 , or
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Figure 65: (Γ, X 0 ) satisfies (R1), a01 = a1
(ii) G has a separation (H 0 , K 0 ) of order 3 such that a1 ∈ V (H 0 ∩ K 0 ), x1 ∈ V (K 0 ),
{a1 , a2 , b1 , b2 } ⊆ V (H 0 ).
We first claim that we may assume there are 4 disjoint paths from a1 to x1 , b1 to b01 ,
a2 to a02 , and b2 to b02 in G. Otherwise, there is some separation (G1 , G2 ) of G of order
at most 3 with {x1 , b1 , a2 , b2 } ⊆ V (G1 ), V (K) ⊆ V (G2 ), and V (K) 6= V (G2 ). Letting
G02 = G2 ∪ {a1 }, we see that (G1 , G02 ) is a separation of G. By the maximality of K, this
separation (G1 , G02 ) is not of order 4. Hence it is of order at most 3, and (Γ, X) ∈ O3 , a
contradiction.
We may assume that {a1 , a02 } is not a 2-cut in K; otherwise, we have 3-cuts {a1 , a02 , b01 }
and {a1 , a02 , b02 } in G demonstrating that (Γ, X) ∈ O3 . Hence there is some path from b01 to
b02 in K − {a1 , a2 }. Let Q be such a path, chosen so that each vertex of Q is cofacial with
a1 . We claim that we may assume there is some xk ∈ X 0 for which there is a path Q0 from
a02 to xk in K − Q. If not, then for each vertex x ∈ X 0 , either x ∈ V (Q), and hence x is
cofacial with a1 , or there is a 2-cut T ⊆ V (Q) in K − a1 separating x from {b01 , a02 , b02 }. But
this again gives (Γ, X) ∈ O3 .
Now the four disjoint paths in H together with Q and Q0 demonstrate that X is feasible
in Γ, a contradiction.
141
(1.2)
Suppose the conditions of (1.1) do not hold.
Since x1 ∈
/ V (K), and by our choice of K, x1 ∈
/ V (E1 ) ∩ V (E2 ). If x1 ∈ V (E1 ∪ E2 ),
assume by symmetry that x1 ∈ V (E1 ).
If there is some 2-cut T in H separating x1 from b1 , a2 , b2 , we note by (3, {a1 , b1 , a2 , b2 })connectivity of G and our choice of K that T ∩V (E1 ∪E2 ) 6= ∅ and T * V (Ei ) for i ∈ {1, 2}.
In this case, we assume by symmetry that T ∩ V (E1 ) 6= ∅.
We claim there are disjoint paths P1 from b1 to b01 , Q from a2 to x1 , and P2 from b2 to
a02 in H − {a1 , b02 }. Otherwise, there is some 2-cut T 0 of H − {a1 , b02 } separating {b1 , a2 , b2 }
from {b01 , x1 , a02 }. So either T 0 ∪ {a1 } is a 3-cut in G, demonstrating that (Γ, X) ∈ O3 , or
T 0 ∪ {a1 , b02 } is a 4-cut in G, demonstrating that (Γ, X) satisfies (R1).
So we have the path Q ∈ P(Γ,X) from a2 to x1 , with P1 ∪ P2 ∪ E1 [a02 , b01 ] ⊆ LQ . We
may assume there is some x2 ∈ X 0 with x2 ∈
/ V (E1 ); otherwise (Γ, X 0 ) ∈ O1 . Since X is
infeasible in Γ, there is no path from a1 to x2 in K − V (E1 ). So there must be some 2-cut
T 0 ⊆ V (E1 ) separating x2 from a2 in K. But then T 0 is a 2-cut in G, and this contradicts
the (3, {a1 , b1 , a2 , b2 })-connectivity of G.
Case 2
Suppose {a1 , a2 } ∩ V (H ∩ K) 6= ∅.
We assume by symmetry that b02 ∈ C[b2 , a1 ], and either b02 = b2 or a01 ∈ C(b02 , a1 ). By
our choice of (H, K) with K maximal, we may find the following internally disjoint paths
in G: P1 from a1 to a01 ; P2 from a2 to a02 ; Q1 and Q2 each from b1 to b01 ; and C[b2 , b02 ], as
shown in Figure 66. Choose such paths so that each vertex of P1 ∪ P2 is cofacial with some
vertex of C[b2 , b02 ], each vertex of Q1 is cofacial with some vertex of P1 , and each vertex of
Q2 is cofacial with some vertex of P2 .
(2.1)
Suppose x1 ∈ V (Pi ), for some i ∈ {1, 2}.
Note that if a01 ∈ V (C), we have P1 = C[a01 , a1 ]. Since x1 ∈
/ V (C), we may assume either
i = 2 or a01 ∈
/ V (C).
Let E1 be the finite facial cycle of G shared by a0i and b01 , and let E2 be the finite facial
cycle shared by a0i and b02 . If x1 ∈ V (Ej ), there is some x2 ∈ X 0 with x2 ∈
/ V (Ej ); otherwise
(Γ, X) ∈ O1 . Note that {b01 , b02 } does not separate a03−i from X in K; otherwise (Γ, X 0 ) ∈ O3 .
142
a
1
11
00
00
11
00
11
P1
P1
00
a01 11
00
11
00
11
Q1
a01
a
1
11
00
00
11
00
11
Q1
11
00
11
00
11
00
00
11
b01
11
00
11
b2 00
1
0
0
1
11
00
11
00
b1
a02
P2
1
0
0
1
11
00
11
b02 00
1
0
0
1
b2
Q2
00
11
11
00
00
11
(a) b02 = b2
b1
11
00
00
11
1
0
0
1
a02
Q2
P2
a2
b01
00
11
11
00
00
11
a2
(b) a01 ∈ C(b02 , a1 )
Figure 66: Disjoint paths in H when {a1 , a2 } ∩ V (K) = ∅
So we find disjoint paths P3 from a03−i to x2 and Q3 from b02 to b01 in K ∪ E1 ∪ E2 , each
disjoint from Pi (Figure 67). Together with Pi [ai , x1 ], P3−i , and Q1 (or Q2 ), these show X
is feasible in Γ, a contradiction.
(2.2)
Suppose x1 ∈
/ V (Q3−i ) for some i ∈ {1, 2}.
We may choose a path P 0 from a1 to x1 in G, disjoint from P3−j ∪ Q3−j ∪ C[b2 , b02 ].
As in the previous case, let E1 be the finite facial cycle of G shared by a0i and b01 , and
let E2 be the facial cycle shared by a0i and b02 . If x1 ∈ V (Ej ), there is some x2 ∈ X 0 with
x2 ∈
/ V (Ej ); otherwise (Γ, X) ∈ O1 . Note that {b01 , b02 } does not separate a03−i from X in
K; otherwise (Γ, X 0 ) ∈ O3 . So we find disjoint paths P3 from a03−i to x2 and Q3 from b02 to
b01 in K ∪ E1 ∪ E2 , each disjoint from P 0 (Figure 68). Together with P 0 , P3−i , and Q3−i ,
these show X is feasible in Γ, a contradiction.
143
a
1
11
00
00
11
00
11
P1
P1
00
a01 11
00
11
00
11
a01
Q1
x111
00
00
11
1
0
0
1
0
1
11
00
11
b2 00
11
00
00
11
00
11
x2
11
00
11
00
b01
1
0
0
1
11
00
11
00
1
0
0
1
a02
P2
a
1
11
00
00
11
00
11
b1
11
00
11
b02 00
Q2
b2
x2
11
00
11
00
1
0
0
1
1
0
0
1
11
00
11
00
a02
x1
11
00
00
11
11
00
00
11
00
11
P2
a2
(a) x1 ∈ V (P1 )
Q2
11
00
00
11
00
11
a2
(b) x1 ∈ V (P2 )
Figure 67: When x1 ∈ V (Pi )
a
01
1
1
0
0
1
a01
x1
1
0
0
1
11
00
11
00
00
11
11
00
00
11
b2
b01
1
0
0
1
x2
1
0
0
1
b01
11
00
11
00
a02
P2
11
00
11
00
b1
Q2
11
00
11
00
a2
Figure 68: When x1 ∈ V (Q1 )
144
b1
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