An Excluded Grid Theorem for Digraphs with Forbidden Minors
Ken-ichi Kawarabayashi
National Institute of Informatics,
JST, ERATO, Kawarabayashi Project
k [email protected]
Abstract
The excluded grid theorem, originally proved by
Robertson and Seymour in Graph Minors V, is one of
the most central results in the study of graph minors. It
has found numerous applications in algorithmic graph
structure theory, for instance as the basis for bidimensionality theory on graph classes excluding a fixed minor.
In 1997, Reed [22] and later Johnson, Robertson,
Seymour and Thomas [16] conjectured an analogous
theorem for directed graphs, i.e. the existence of a
function f : N → N such that every digraph of directed
tree-width at least f (k) contains a directed grid of order
k. In an unpublished manuscript from 2001, Johnson,
Robertson, Seymour and Thomas gave a proof of this
conjecture for planar digraphs but no result beyond
planar graphs is known to date.
In this paper we prove the conjecture for the case
of digraphs excluding a fixed undirected graph as a minor. For algorithmic applications our theorem is particularly interesting as it covers those classes of digraphs to
which, on undirected graphs, theories based on the excluded grid theorem such as bidimensionality theory apply. We expect similar applications for directed graphs
in particular to algorithmic versions of Erdős-Pósa type
results and the directed disjoint paths problem.
1
Introduction
Structural graph theory has proved to be a powerful tool
for coping with computational intractability. It provides
a wealth of concepts and results that can be used to design efficient algorithms for hard computational problems on specific classes of graphs occurring naturally in
applications. Of particular importance is the concept
of tree-width, introduced by Robertson and Seymour as
part of their seminal graph minor series [26]. Graphs
of small tree-width can recursively be decomposed into
subgraphs of constant size which can be combined in
a tree-like way to yield the original graph. This property allows to use algorithmic techniques such as dynamic programming to solve many hard computational
Stephan Kreutzer
Technical University Berlin
[email protected]
problems efficiently on graphs of small tree-width. In
this way, a huge number of problems has been shown
to become tractable, e.g. solvable in linear or polynomial time, on graph classes of bounded tree-width. See
e.g. [3, 2, 4, 12] and references therein. But methods
from structural graph theory, especially graph minor
theory, also provide a powerful and vast toolkit of concepts and ideas to handle graphs of large tree-width and
to understand their structure.
One of the most fundamental theorems in this context is the excluded grid theorem, proved by Robertson
and Seymour in [30]. It states that there is a function
f : N → N such that every graph of tree-with at least
f (k) contains a k ×k-grid as a minor. This function, initially being enormous, has subsequently been improved
and is now single exponential [18, 29]. Only recently,
Chekuri and Chuzhoy announced a polynomial dependency of the size of the grid with respect to its treewidth [5].
The excluded grid theorem is important for structural graph theory as well as for algorithmic applications. For instance, algorithmically it is the basis
of an algorithm design principle called bidimensionality theory, which has been used to obtain many approximation algorithms, PTASs, subexponential algorithms and fixed-parameter algorithms on graph classes
excluding a fixed minor. These include feedback vertex set, vertex cover, minimum maximal matching, face
cover, a series of vertex-removal parameters, dominating set, edge dominating set, R-dominating set, connected dominating set, connected edge dominating set,
connected R-dominating set and unweighted TSP tour.
See [7, 8, 9, 10, 15, 14] and references therein.
Furthermore, the excluded grid theorem also plays
a key rôle in Robertson and Seymour’s graph minor algorithm and their solution to the disjoint paths problem
[27] in a technique known as the irrelevant vertex technique. Here, a problem is solved by showing that it can
be solved efficiently on graphs of small tree-width and
otherwise, i.e. if the tree-width is large and therefore the
graph contains a large grid, that a vertex deep in the
middle of the grid is irrelevant for the problem solution and can therefore be deleted. This yields a natural
recursion that eventually leads to the case of small treewidth. This technique has been particularly successful
on classes of graphs excluding a fixed minor, the objects of study in this paper, as on such classes one can
show that the grid forms an almost planar subgraph (see
Theorem 6.1). See [1] for an application of the irrelevant
vertex technique to disjoint paths on planar graphs.
In terms of graph structural aspects, the excluded
grid theorem is the basis of the more advanced structure
theorems in graph minor theory such as in [27] (see
Theorem 6.1) and [28].
The structural parameters and techniques discussed
above all relate to undirected graphs. However, in
many applications in computer science, directed graphs
are a much more natural model. Given the enormous
success width parameters had for problems defined on
undirected graphs, it is natural to ask whether they
can also be used to analyse the complexity of NP-hard
problems on digraphs. While in principle it is possible
to apply the structure theory for undirected graphs
to directed graphs by ignoring the direction of edges,
this implies a significant information loss. Hence, for
computational problems whose instances are directed
graphs, methods based on the structure theory for
undirected graphs may be less useful.
As a first step towards a structure theory specifically for directed graphs, Reed [23] and Johnson,
Robertson, Seymour and Thomas [16] proposed a concept of directed tree-width and showed that the kdisjoint paths problem is solvable in polynomial time for
any fixed k on any class of graphs of bounded directed
tree-width [16]. Reed and Johnson et al. also conjecture
a directed analogue of the excluded grid theorem.
the basis of more general structure theorems. Furthermore, it is likely that the duality of directed tree-width
and excluded directed grids will make it possible to develop algorithm design techniques such as bidimensionality theory or the irrelevant vertex technique for directed graphs. We are particularly optimistic that this
approach would prove very useful for algorithmic versions of Erdős-Pósa type results and in the study of
the directed disjoint paths problem. In a recent breakthrough, Cygan et al. [6] showed that the planar directed
disjoint paths problem is fixed-parameter tractable using an irrelevant vertex technique (but based on a different type of directed grid). They show that if a planar digraph contains a a grid-like subgraph of sufficient
size, then one can delete a vertex in this grid without
changing the solution. The bulk of the paper then analyses what happens if such a grid is not present. If one
could prove a similar irrelevant vertex rule for the directed grids used in Reed and Johnson et al.’s conjecture, then the excluded grid conjecture would immediately yield the dual notion in terms of directed treewidth for free. The directed disjoint paths problem beyond planar graphs therefore is a prime algorithmic application we envisage for directed grids.
Another obvious application of the excluded grid
conjecture is to Erdős-Pósa type results such as
Younger’s conjecture proved by Reed et al. in 1996 [24].
In fact, in their proof of Younger’s conjecture, Reed
et al. construct a kind of a directed grid. This technique was indeed a primary motivation for considering
directed tree-width and a directed grid minor as a proof
of the directed excluded grid conjecture would yield a
simple proof for Younger’s conjecture.
In this paper we prove a significant step towards
a proof of the conjecture by establishing an excluded
grid theorem for digraphs excluding a fixed graph
Conjecture 1.1. ([22, 16]) Every digraph of high di- as undirected minor. More precisely, we prove the
rected tree-width contains a large directed grid, i.e. there following theorem.
is a function f : N → N such that every digraph of directed tree-width at least f (k) contains a cyclic grid of Theorem 1.2. Let C be a class of directed graphs exorder k as a butterfly minor
cluding a fixed undirected graph H as a minor. There
is a computable function f : N → N such that for all
Here, a directed grid consists of k concentric directed G ∈ C and all k ∈ N, if the directed tree-width of G is
cycles and 2k paths connecting the cycles in alternat- at least f (k) then G contains a cylindrical grid of order
ing directions. See Section 3 for details and see Defini- k.
tion 2.1 for a definition of butterfly minors.
In an unpublished manuscript, Johnson et al. [17]
See Section 3 for a formal definition of a cylindriproved the conjecture for planar digraphs. However, cal grid. Our result significantly extends the result in
despite the conjecture being open for 15 years now, no [17] for the case of planar digraphs. The techniques we
progress beyond the planar case has been made to date. develop in this paper are quite different from the techSuch a grid theorem for digraphs would be an im- niques used in [17] and we are optimistic that our apportant step towards a more general structure theory for proach will eventually allow us to establish the directed
directed graphs based on directed tree-width, similar to excluded grid theorem in full generality. However, much
the excluded grid theorem for undirected graphs being more research and significant new ideas are needed to
achieve this goal.
As far as algorithmic applications are concerned,
our result already covers many important cases as graph
classes excluding a minor are exactly the classes of
graphs to which (undirected) bidimensionality theory
applies which we hope to generalise to digraphs. Also,
it covers the most promising cases for solutions to
the directed disjoint paths problem, such as digraphs
of bounded genus etc. In this paper, however, we
focus on the structural aspects and prove the excluded
grid theorem for H-minor free digraphs. We leave
applications for future research.
abstract graph of G. A minor H ⊆ G is a minor of the
abstract graph G0 . Besides undirected minors we also
use a directed version of the minor relation. There is no
agreed concept of digraph minors and we therefore use
a very weak version, called butterfly minors (see [16]).
Definition 2.1. (butterfly minor) Let G be a digraph. An edge e = (u, v) ∈ E(G) is butterflycontractible if e is the only outgoing edge of u or the only
incoming edge of v. In this case the graph G0 obtained
from G by butterfly-contracting e is the graph with vertex set V (G) − {u, v} ∪ {xu,v }, where xu,v is a fresh
vertex. The edges of G0 are the same as the edges of G
Organisation and high level overview of the proof
except for the edges incident with u or v. Instead, the
structure. We review basic concepts of graph theory
new vertex xu,v has the same neighbours as u and v,
and combinatorics in Section 2. In Section 3 we state
eliminating parallel edges. A digraph H is a butterflyour main result and present relevant definitions. In
minor of G if it can be obtained from a subgraph of G
Sections 4-7, then, we present the proof of our main
by butterfly contraction.
result. On a high level, the proof works as follows. If
a digraph G has high directed tree-width, it contains a
Definition 2.2. (intersection graph) Let P and
directed bramble of very high order (see Section 3). From
Q be sets of pairwise disjoint paths in a digraph G. The
this bramble we construct a cylindrical grid in several
intersection graph I(P, Q) of P and Q is the bipartite
stages. We first show that if we have a large bramble we
graph with vertex set P ∪ Q and an edge between P ∈ P
also have a large path system, a collection of paths such
and Q ∈ Q if P ∩ Q 6= ∅.
that between any two of them there is a huge linkage
in both directions. We then try to get the paths in the
We need the following lemma from [11].
system and the linkages pairwise disjoint (see Section 4).
If this succeeds, we have a huge undirected clique-minor, Lemma 2.3. ([11]) Let G be a bipartite graph with
which is impossible. Hence, this process must fail and bipartition (A, B), |A| = a, |B| = b, and let c ≤ a
as a result we get two huge linkages which pairwise and d ≤ b be positive integers. Assume that G has at
intersect heavily. This is called a web (see Section 4). most (a − c)(b − d) edges. Then there exist C ⊆ A and
In Section 5 we show that this web can be ordered and D ⊆ B such that |C| = c and |D| = d and C ∪ D is
rerouted to obtain a nicer version of a web called a fence. independent in G.
A fence is essentially a cylindrical grid with one edge of
each cycle deleted. Hence, to obtain the cylindrical grid,
We also need the next result by Erdős and Szekall that is needed is to find a linkage from the bottom eres [13].
of the fence back to its top that is disjoint from the
fence. This, however, is by far the most difficult part Theorem 2.4. Let s, t be integers, and let n = (s −
of the proof, which we present in Section 7. It is in 1)(t−1)+1, and let a1 , . . . , an be distinct integers. Then
this section, that we crucially use the fact that we work there exist 1 ≤ i1 < · · · < is ≤ n s.t. ai < · · · < ai , or
1
s
in digraphs excluding a minor. This allows us to use a there exist 1 ≤ i1 < · · · < it ≤ n s.t. ai > · · · > ai .
1
t
structure theorem by Robertson and Seymour which we
present in Section 6. We conclude the proof and state 2.2 Minimal Linkages. A linkage P is a set of
open problems in Section 8.
mutually vertex-disjoint directed paths (dipaths) in a
Due to space restrictions we defer the proof of most digraph. For two vertex sets Z1 and Z2 , P is a Z1 of our results to the full version of this paper.
Z2 linkage if each member is a dipath from a vertex
in Z1 to some other vertex in Z2 . The order of the
linkage, denoted by |P|, is the number of paths. In
2.1 Background from graph theory. For any set slightly informal notation, we will sometimes identify a
U and k ∈ N we define [U ]≤k := {X ⊆ U : |X| ≤ k}. linkage P with the subgraph consisting of the paths in
We define [U ]=k etc. analogously. Let G be a digraph. P.
We refer to its vertex set by V (G) and its edge set by
Let G be a digraph and A ⊆ V (G). A is well-linked,
E(G). We refer to the underlying undirected graph G0 if for all X, Y ⊆ A with |X| = |Y | = r there is an X −Y obtained from G by ignoring the direction of edges as the linkage of order r.
2
Preliminaries
of digraphs excluding a fixed undirected minor. We
will prove this result based on the concept of directed
brambles which is equivalent to directed tree-width. Let
us first recall the definition of directed brambles and
cyclindrical grid as defined in [22, 23, 16].
paths in the obvious way. Let a cylindrical wall of order
k be the cylindrical grid of order k where each vertex u
of degree 4 is replaced by an edge (u1 , u2 ), where u1 , u2
are fresh vertices, so that u1 has the in-neighbours of
u and u2 the out-neighbours of u. Then we prove that
any H-minor free digraph with a bramble of order f (k)
Definition 3.1. (cylindrical grid) A cylindrical contains a subdivision of the cylindrical wall of order k.
grid of order k, for some k ≥ 1, is a digraph Gk
comprising k directed cycles C1 , . . . , Ck , pairwise vertex 4 Getting a web
disjoint, together with a set of 2k pairwise vertex The main objective of this section is to show that every
disjoint paths P1 , . . . , P2k such that
digraph containing a bramble of high order either con• each path Pi has exactly one vertex in common with tains a large clique as an undirected minor or contains
a structure that we call a web.
each cycle Cj ,
• the paths P1 , . . . , P2k appear on each Ci in this
order
• for odd i the cycles C1 , . . . , Ck occur on all Pi in
this order and for even i they occur in reverse order
Ck , . . . , C1 .
See Figure 1 for an illustration of G4 .
Definition 3.2. Let G be a digraph. A bramble in G
is a set B of strongly connected subgraphs B ⊆ G such
that if B, B 0 ∈ B then B ∩ B 0 6= ∅ or there are edges e, e0
such that e links B to B 0 and e0 links B 0 to B.
A cover of B is a set X ⊆ V (G) of vertices such
that V (B) ∩ X 6= ∅ for all B ∈ B. Finally, the order
of a bramble is the minimum size of a cover of B. The
bramble number bn(G) of G is the maximum order of
a bramble in G.
Definition 4.1. ((p, q)-web) Let p, q, d ≥ 0 be integers. A (p, q)-web with avoidance d in a digraph G
consists of two linkages P = {P1 , . . . , Pp } and Q =
{Q1 , . . . , Qq } such that
1. P is an A−B linkage for two distinct vertex sets
A, B ⊆ V (G) and Q is a C−D linkage for two
distinct vertex sets C, D ⊆ V (G),
2. for 1 ≤ i ≤ q, Qi intersects all but at most
paths in P and
1
d
·p
3. P is Q-minimal.
We say that (P, Q) has avoidance d = 0 if Qi intersects
all paths in P, for all 1 ≤ i ≤ q.
The set C ∩ V(Q) is called the top of the web,
denoted top
(P, Q) , and D ∩ V (Q) is the bottom
bot (P, Q) . The web (P, Q) is well-linked if C ∪ D
is well-linked.
The next lemma is mentioned in [23] and can be
proved by converting brambles into havens and back
The notion of top and bottom refers to the intuition,
using [16, (3.2)].
used in the rest of the paper, that the paths in Q are
Lemma 3.3. There are constants c, c0 such that for all thought of as vertical paths and the paths in P as
horizontal. In this section we will prove the following
digraphs G, bn(G) ≤ c · dtw(G) ≤ c0 · bn(G).
theorem.
Using this lemma we can state our main theorem
equivalently as follows, which is the result we prove in Theorem 4.2. For every k, p, l, c ≥ 1 there is an
integer l0 such that the following holds. Let G be a
this paper.
digraph of bramble number at least l0 . Then either G
Theorem 3.4. Let C be a class of directed graphs ex- contains a Kk as an undirected minor or a (p0 , l · p0 )cluding a fixed undirected graph H as a minor. There web with avoidance c, for some p0 ≥ p, such that the top
is a computable function f : N → N such that for all and the bottom of the web are elements of a well-linked
G ∈ C and all k ∈ N, if G contains a bramble of order set A ⊆ V (G).
at least f (k) then G contains a cylindrical grid of order
The starting point for proving the theorem are
k as a butterfly minor.
brambles of high order in directed graphs. In the first
The previous theorem can also be restated in terms step we adapt an approach developed in [25] to our
of the more canonical notion of subdivisions, where a setting. We will now define the first of various subsubdivision of a digraph G is a digraph obtained from structures we are guaranteed to find in graphs of high
G by replacing edges by pairwise vertex disjoint directed directed tree-width.
Definition 4.3. (path system) Let G be a digraph
3.
and let l, p ≥ 1. An l-linked path system of order p
is a sequence P := (P1 , . . . , Pp ) of paths such that each
contains l elements of a well-linked set A.
4.
The system P is clean if for all 1 ≤ i < j ≤ p there
is a linkage Li,j from Pi to Pj of order l such that for
all 1 ≤ i < j ≤ p and all Q ∈ Li,j , Q ∩ Ps = ∅ for all
1 ≤ s ≤ p with s 6∈ {i, j}.
The
For 1 ≤ j ≤ q, the paths P1 ∩Qj , . . . , P2p ∩Qj are in
order in Qj , and the first vertex of Qj is in V (P1 )
and the last vertex is in V (P2p ).
For 1 ≤ i ≤ 2p, if i is odd then Pi ∩ Q1 , . . . , Pi ∩ Qq
are in order in Pi , and if i is even then Pi ∩
Qq , . . . , Pi ∩ Q1 are in order in Pi .
fence F is well-linked if A ∪ B is well-linked.
The main theorem of this section is to show that
The proof of the next lemma and the various
any
digraph with a large web where bottom and top
intermediate lemmas needed for its proof is deferred to
come
from a well-linked set contains a large well-linked
the full version of this paper.
fence.
Lemma 4.4. Let G be a digraph. There is a computable
0
function f3 : N4 → N such that for all integers l, p, k, c ≥ Theorem 5.2. For every p, q ≥ 1 there is a p such
0 0
1, if G contains a bramble of order f3 (l, p, k, c) then G that any digraph G containing a well-linked (p , p )-web
contains a clean l-linked path system P of order p or a contains a well-linked (p, q)-fence.
well-linked (p, k · p)-web of avoidance c.
To prove the previous theorem we first establish a
weaker
version where instead of a fence we obtain an
The following lemma completes the proof of Theoacyclic
grid.
We give the definition first.
rem 4.2.
Definition 5.3. (acyclic grid) An acyclic (p, q)Lemma 4.5. For every k, p, l, c ≥ 1 there is an integer l0
grid is a (p, q)-web P = {P1 , . . . , Pp }, Q =
such that the following holds. Let P be a clean l0 -linked
{Q1 , . . . , Qq } with avoidance d = 0 such that
path system of order k. Then either G contains a Kk as
an undirected minor or a (p0 , l · p0 )-web with avoidance
1. for 1 ≤ i ≤ p and 1 ≤ j ≤ q, Pi ∩ Qj is a path Rij ,
c, for some p0 ≥ p, such that the top and the bottom of
2. for 1 ≤ i ≤ p, the paths Ri1 , . . . , Riq are in order
the web are elements of a well-linked set A ⊆ V (G).
in Pi , and
We close the section with a simple lemma allowing
3. for 1 ≤ j ≤ q, the paths R1j , . . . , Rpj are in order
us to reduce every web to a web with avoidance 0.
in Q .
j
0
0
d
0
d−1 p
Lemma 4.6. Let p , q , d be integers and let p ≥
and q ≥ q 0 · 1pp . If a digraph G contains a (p, q)-web
d
(P, Q) with avoidance d then it contains a (p0 , q 0 )-web
with avoidance 0.
5
From Webs to Fences
The definition of top and bottom is taken over from
the underlying web.
Theorem 5.4. For all integers t, d ≥ 1, there is an
integer p such that every digraph G containing a welllinked (p, p)-web with avoidance d contains a well-linked
acyclic (t, t)-grid.
The objective of this section is to show that if a digraph
Theorem 5.2 is now easily obtained from Theocontains a large web, then it also contains a big fence
rem
5.4 using the following lemma, which is (4.7) in
whose bottom and top come from a well-linked set. We
[24].
It is easily seen that the top and bottom of the
give a precise definition of a fence and then state our
fence
are subsets of the top and bottom of the acyclic
main theorem.
grid it is constructed from.
Definition 5.1. (fence) Let p, q be integers.
A
5.5. For every integer p ≥ 1, there is an integer
(p, q)-fence in a digraph G is a sequence F := Lemma
00
≥
1
such
that every digraph with a (p00 , p00 )-grid has
p
(P1 , . . . , P2p , Q1 , . . . , Qq ) with the following properties:
a (p, p)-fence such that the top and bottom of the fence
1. P1 , . . . , P2p are mutually disjoint paths of G and are subsets of the top and bottom of the grid.
{Q1 , . . . , Qq } is an A-B-linkage for two distinct
So far we have shown that every digraph which
sets A, B ⊆ V (G), called the top
(A) and bottom
contains
a large well-linked web also contains a large
(B) of the fence, denoted top F and bot F .
well-linked fence (P, Q). The well-linkedness of (P, Q)
2. For 1 ≤ i ≤ 2p and 1 ≤ j ≤ q, Pi ∩ Qj is a path implies the existence of a minimal bottom-up linkage as
(and therefore non-empty).
defined in the following definition.
Definition 5.6. Let (P, Q) be a fence. A (P, Q)bottom-up linkage is a linkage R from bot(P, Q) to
top(P, Q). It is called minimal (P, Q)-bottum-up linkage, if R is (P, Q)-minimal.
s
s
s
s
s
s
s
s
We close this section by establishing a few simple
routing principles in fences which will be needed below.
The first is (3.2) in [24].
Lemma 5.7. Let (P1 , . . . , P2p , Q1 , . . . , Qq ) be a (p, q)fence in a digraph G, with the top A and the bottom
B. Let A0 ⊆ A and B 0 ⊆ B with |A0 | = |B 0 | =
r for some r ≤ p.S Then thereSare vertex disjoint
paths Q01 , . . . , Q0r in 1≤i≤2p Pi ∪ 1≤j≤q Qj such that
(P1 , . . . , P2p , Q01 , . . . , Q0r ) is a (p, r)-fence with top A0
and bottom B 0 .
The next lemma establishes another routing property of fences with bottom-up linkages. The second part
of the lemma will be very useful below.
Lemma 5.8. Let W be a (t, t)-fence, for some t ≥ 0,
and let R be a W-bottom-up linkage of order t. For
any t0 ≤ 2t there are linkages P 0 and Q0 so that (P 0 , Q0 )
forms a (t0 , t0 )-sub-fence W 0 ⊆ W such that there are
t0 /2 disjoint paths from bot(W 0 ) to top(W 0 ).
Moreover, if there is a linkage R0 ⊂ R of order t0
such that no path in R0 intersects any path in W 0 , then
the above t0 /2 paths can be chosen to be disjoint from
W 0 except for the endvertices.
Proof. Let us take t0 disjoint paths in R from some F 0 ⊂
bot(W) to some H 0 ⊂ top(W). It is straightforward to
choose a subfence W 0 ⊂ W so that there are t0 disjoint
paths from H 0 to top(W 0 ) and t0 disjoint paths from
bot(W 0 ) to F 0 such that these 2t0 paths have no inner
vertex in W 0 . Together with the linkage R, this yields a
half-integral linkage of order t0 from bot(W 0 ) to top(W 0 )
and thus, by Lemma 2.6, an integral linkage of order
t0 /2. So, the claim holds.
The last conclusion follows if we only consider the
linkage R0 instead of R.
6
Flat walls in Kt -minor-free graphs
The arguments so far applied to general digraphs and
did not assume any properties of digraphs excluding a
fixed undirected minor. In the next section, however, we
will make use of this fact. In this section, therefore, we
recall a structure theorem for graphs excluding a fixed
minor proved by Robertson and Seymour, which we will
use towards the end of our proof.
An elementary wall of height eight is depicted in
Figure 2. An elementary wall of height h for h ≥ 2
is similar. It consists of h levels each containing h
s
s
s
s
s
s
s
s
Figure 2: An elementary wall of height 8
s s s s s
s
s
s
s s s s
s
s
s s s s s s s
s
s
s
ss ss
s s
s
s
s
s
s
s
s s s
s ss s
s
s
sssss
s s s
s
Figure 3: A wall of height 3
bricks, where a brick is a cycle of length six. A wall
of height h is obtained from an elementary wall of
height h by subdividing some of the edges, i.e. replacing
the edges with internally vertex-disjoint paths with the
same endpoints (see Figure 3). The nails of a wall are
the vertices of degree three within it. Any wall has a
unique planar embedding. The perimeter of a wall W ,
denoted per(W ), is the boundary of the unique face in
this embedding which contains more than six nails.
In the remainder of this section, we shall mention a
structural result by Robertson and Seymour concerning
graphs which have a large wall but no large clique minor.
To state this result we will need a few definitions.
For any wall W in a given graph G, there is a unique
component U of G − per(W ) containing W − per(W ).
The compass of W , denoted comp(W ), is the subgraph
of G induced by V (U ) ∪ V (per(W )). A subwall of a wall
W is a wall which is a subgraph of W . A subwall of W
of height h is proper if it consists of h consecutive bricks
from each of h consecutive rows of W . The exterior of
a proper subwall W 0 of a wall W is W − W 0 . We say
a proper subwall W 0 is dividing in G if the compass of
W 0 in G is disjoint from W − W 0 .
A wall is flat if its compass does not contain two
vertex-disjoint paths connecting the diagonally opposite
corners. Note that if the compass of W has a planar embedding whose infinite face is bounded by the perimeter
of W then W is clearly flat. In order to characterize flat
walls, we use the result of Seymour [31], Thomassen [32],
and others on the two disjoint paths problem.
By the characterization of graphs for the two disjoint paths problem (see [31, Theorem 4.1] for example),
a wall W is flat if and only if there are pairwise disjoint
sets A1 , . . . , Al ⊆ V (comp(W )) containing no corners of
W such that
(1) for 1 ≤ i, j ≤ l with i 6= j, N (Ai ) ∩ Aj = ∅,
Definition 7.2. (generalised quasi-mesh) Let
p, q, t0 , t be integers, and let r1 , r2 ≥ 0 be integers with
r2 + r1 ≤ t0 and t ≤ t0 . A generalised (p, q, t0 , t, r1 , r2 )quasi-mesh in a digraph G consists of a (p, q)-grid
P = (P1 , . . . , Pp ), Q = (Q1 , . . . , Qq ) together with the
linkage R = (R1 , . . . , Rt0 ) such that
(2) for 1 ≤ i ≤ l, |N (Ai )| ≤ 3, and
1. for 1 ≤ i ≤ t0 , Ri intersects all paths in P ∪ Q,
(3) if W 0 is the graph obtained from comp(W ) by
deleting Ai and adding new edges joining every pair
of distinct vertices in N (Ai ) for each i, then W 0 can
be drawn in a plane so that all corners of W are on
the outer face boundary.
2. for 1 ≤ i ≤ r1 and for 1 ≤ j, j 0 ≤ p with j 6= j 0 ,
if v1 , v2 ∈ V (Ri ∩ Pj ) and v0 ∈ V (Ri ∩ Pj 0 ) then
v0 does not lie in Ri between v1 and v2 . Similarly,
for 1 ≤ i ≤ r1 and for 1 ≤ j ≤ p and 1 ≤ j 0 ≤ q,
if v1 , v2 ∈ V (Ri ∩ Pj ) and v0 ∈ V ((Ri ∩ Qj 0 ) − Pj )
then v0 does not lie in Ri between v1 and v2 .
Robertson and Seymour (Theorem (9.6) in [27]) proved
the following algorithmic result (see also [19]).
Theorem 6.1. For any t ≥ 0 and h ≥ 2, there is a
computable constant f (t, h) such that the following can
be done in O(m) time, where m is the number of edges
of a given graph G.
Input: A graph G, a wall H of height at least f (t, h).
Output: Either
1. Kt -minor, or
2. a subset X ⊆ V (G) of order at most 2t and a
proper subwall of height h that is dividing and flat
in G − X. In addition, a flat embedding of it is also
given.
7 Constructing a cylindrical grid
In this section we prove the following result which,
combined with the results in the previous sections,
completes the proof of Theorem 3.4 and hence of
Theorem 1.2.
3. for 1 ≤ i ≤ r1 and for 1 ≤ j, j 0 ≤ q with j 6= j 0 ,
if v1 , v2 ∈ V (Ri ∩ Qj ) and v0 ∈ V (Ri ∩ Qj 0 ) then
v0 does not lie in Ri between v1 and v2 . Similarly,
for 1 ≤ i ≤ r1 and for 1 ≤ j ≤ q and 1 ≤ j 0 ≤ p,
if v1 , v2 ∈ V (Ri ∩ Qj ) and v0 ∈ V ((Ri ∩ Pj 0 ) − Qj )
then v0 does not lie in Ri between v1 and v2 .
4. for r1 + 1 ≤ i, i0 ≤ r2 with i 6= i0 and for 1 ≤ j ≤ p,
if v1 , v2 ∈ V (Ri ∩ Pj ), then there is no vertex
v0 ∈ V (Ri0 ∩ Pj ) that lies in Pj between v1 and
v2 .
5. for r1 + 1 ≤ i, i0 ≤ r2 with i 6= i0 and for 1 ≤ j ≤ q,
if v1 , v2 ∈ V (Ri ∩ Qj ), then there is no vertex
v0 ∈ V (Ri0 ∩ Qj ) that lies in Qj between v1 and
v2 .
6. For any edge e ∈ E(R − (P ∪ Q)), G − e has no t00
disjoint paths from bot(P, Q) to top(P, Q), where
t00 ≤ t is the number of paths in R from bot(P, Q)
to top(P, Q).
Theorem 7.1. For every graph H of order h and for
Moreover, if R0 ∈ R is not a path from bot(P, Q)
all p0 ≥ 0 there are constants p, q, t ≥ 0 such that if a
to top(P, Q), then for any edge e = r0 r00 ∈ E(R0 ),
digraph G whose abstract graph does not contain H as
there are at most t vertex-disjoint paths from P 0 to
a minor contains a (p, q)-fence (P, Q) and a minimal
P 00 , where P 0 ∪ e ∪ P 00 = R and P 0 contains r0 .
(P, Q)-bottom-up linkage R of order t, then G contains
a cylindrical grid of order p0 .
Lemma 7.3. For all integers r3 and p00 , t0 , t with t0 ≥ t
00 00
Let (P, Q) be a fence and R a bottom-up linkage and r3 ≥ 4t there are p, q ≥ 00 with p + q ≥ r3 p (p + t)
for
as in the statement of the theorem. We first need a such that if there is a (p, q,0 t ,0t, r1 , r2 00)-quasi-mesh,
0
0
,
q
with
p
=
p
+
q
,
either
some
r
,
r
,
then
for
some
p
1
2
few technical lemmas. The first lemma uses a structure
called generalised quasi-mesh 1 . In the lemma, we start
with (P, Q) and R as before, but then some paths in R
will be split so that they are no longer bottom-up paths,
which implies some technicalities as in Condition 6 of
the following definition.
1 The
term generalised quasi-mesh is used as they generalise
a structure called quasi-mesh which, in the full version of this
paper, is used in the proof of our main result of Section 5.
1. there is a (p0 , q 0 , t0 , t, r1 + 1, r2 − 1)-quasi-mesh, or
2. there is a (p0 , q 0 , t0 + 1, t, r1 , r2 + 1)-quasi-mesh by
deleting one edge from a path in R, or
3. there is a path R ∈ R, a subpath R0 of R and a
set P 0 ⊆ P ∪ Q of order at least r3 such that R0
consists only of edges in P ∪ Q and hits every path
in P 0 . Moreover, if r0 is the last vertex in R0 and
r00 is the vertex with r0 r00 ∈ E(R), then the subpath
R00 of R from r00 to the endvertex of R hits vertices
of P 0 which are in a grid created by P 0 , and which
are covered by at most 4t paths in P 0 .
applied to a = q/3, b = t/10 and c = a/10, d = b/10,
there are C ⊆ Q̄ and D ⊆ R0 such that C ∪ D either is
independent in I2 or in I¯2 . In the first case, no path in
C intersects any path in D. In the second case, every
path in C intersects every path in D. Let R00 ⊆ R0
Proof of Theorem 7.1.
So far we have argued for correspond to the paths in D and let Q0 ⊆ Q̄ correspond
general digraphs. In this section we will finally use that to the paths in C. We now distinguish between four
the abstract graph of G excludes a graph H as a minor. different cases depending on whether P 0 and Q0 come
Clearly, any (p, q)-fence in G induces a (p, q)-wall in the from I or I¯ and I or I¯ , resp. Due to lack of space,
1
1
2
2
abstract graph. Hence, we can apply Theorem 6.1. Let the proof of all four cases is deferred to the full version
f be the function defined in this theorem. We now take of this paper. Proving these four sub-cases concludes
the following constants:
the proof of Theorem 7.1.
p4 ≥ 72p3 , and p2 ≥ 6(2(p1 − 1)2 + 2),
and take p3 so that p = p3 in Theorem 5.4 implies t = p2
in Theorem 5.4, and moreover p + q = p3 in Lemma 7.3
implies p0 + q 0 = 18p2 in Lemma 7.3. We also assume
p4 ≥ t ≥ p3 .
We take p, q ≥ f (|H|, 2 · p4 ), aiming to construct
a cylindrical grid of order p1 . Suppose we are given
a (p, q)-fence (P, Q) and a minimal (P, Q)-bottom-up
linkage R of order t.
We now apply Theorem 6.1 to the underlying wall.
As there is no H-minor in the abstract graph G we
can delete the vertices of X in Theorem 6.1 and, using
Lemma 5.8, we obtain a flat and dividing (p4 , p4 )-fence
M̄ := (P, Q) with t disjoint paths from the bottom to
the top of M̄ .
For technical reasons we only consider the middle
cylinder of the fence, i.e. only 31 of the paths in Q. More
precisely, if Q1 , . . . , Qq are the paths appearing in this
order in Q (from left to right), we set
(a) Q̄ := {Qq/3+1 , . . . , Q2q/3 } and
(b) we also “shorten” the paths in Q̄ and let each path
in Q̄ start at a vertex in P2p/3+1 and end at a vertex
in P4p/3 .
Let us construct the intersection graph I1 :=
I(P, R) (see Definition 2.2). We also consider the “dual
graph” I¯1 of I1 . Namely, for a vertex x ∈ P and y ∈ R,
xy ∈ E(I¯1 ) if x and y do NOT intersect.
We first consider the graph I1 and its dual graph
I¯1 . By Lemma 2.3 applied to a = 2p/3, b = t and
c = a/10, d = b/10, either I1 has C ⊆ P and D ⊆ R
such that C ∪ D is independent in I1 or I¯1 has C ⊆ P
and D ⊆ R such that C ∪ D is independent in I¯1 . In
the first case, no path in C intersects any path in D.
On the other hand, in the second case, every path in C
intersects every path in D. In either case let R0 ⊆ R
correspond to the paths in D and let P 0 ⊆ P correspond
to the paths in C.
We now consider the intersection graph I2 :=
I(Q̄, R0 ) and its dual I¯2 as above. By Lemma 2.3,
8
Conclusion of the proof and future work
We can now combine the various theorems established
above to prove our main result, Theorem 3.4, and
hence Theorem 1.2, as follows. Let G be a digraph
excluding an undirected minor H of order k. To prove
Theorem 1.2, for every h ≥ 1 we need to find an integer
w so that if G has directed tree-width at least w then
G contains a cylindrical grid of order h. Fix h ≥ 1.
By Theorem 7.1, there are constants p1 , q1 , t1 depending only on k and h so that if G contains a (p1 , q1 )fence (P, Q) and a minimal (P, Q)-bottom-up linkage
R of order t1 , then G contains a cylindrical grid of order h. By Theorem 5.2, there is a constant p2 only
depending on p1 and q = max{q1 , t1 } = q1 such that
if G contains a well-linked (p2 , p2 )-web then it contains
the well-linked (p1 , q1 )-fence (and hence the bottom-up
linkage R) as required. Finally, Theorem 4.2 together
with Lemma 4.6 implies that there is a constant p3 only
depending on k and p2 such that if G contains a bramble of order p3 then G contains the well-linked (p2 , p2 )web as required. Hence, for every h we can choose
p1 , q1 , t1 , p2 and p3 as required by these theorems so
that if G contains a bramble of order p3 then we are
guaranteed a cylindrical grid of order h. This proves
Theorem 3.4. Theorem 1.2 then follows immediately
from Theorem 3.4 using Lemma 3.3.
Future work. In this paper we prove the directed excluded grid conjecture for classes of digraphs excluding a
fixed undirected minor, generalising the previously best
results in [17]. The obvious open problem is to provide
a full proof of the directed excluded grid theorem. We
believe that our approach here shows a possible route
towards proving the general case but much more research and significant new ideas are needed to achieve
this goal.
The second avenue to pursue is to establish algorithmic and structural applications of the excluded grid
theorem proved here, for instance for the directed disjoint paths problem. Another promising application is
the Erdős-Pósa property for directed graphs. For undi-
rected graphs, the grid theorem provides the key to
proving that exactly planar graphs have this property
(see [30]) and it is very likely that the directed grid
theorem would be equally important for this question
on directed graphs. In this paper, however, our focus
was on obtaining the structural result and we leave the
applications to future research.
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