Information
Processing Letters 62 ( 1997) 315-322
A linear-time algorithm for four-partitioning
four-connected planar graphs
Shin-ichi Nakano I, Md. Saidur Rahman *q2,Takao Nishizeki ’
Graduate School of
Information Sciences, Tohoku University, Set&i 980-77. Japan
Received 8 November
1996
Communicated by T. Asano
Abstract
Given a graph G = (YE), four distinct vertices uI.u2,u3&4 E V and four natural numbers nl , n2, n3. ?Q such that
ni = IV\, we wish to find a partition VI, &, 6, & of the vertex set V such that ui E x, 1x1 = ni and L$ induces a
connected subgraph of G for each i. 1 < i < 4. In this paper we give a simple linear-time algorithm to find such a partition
if G is a 4-connected planar graph and ~1. ~2. 143and u4 are located on the same face of a plane embedding of G. Our
algorithm is based on a “4canonical decomposition” of G, which is a generalization of an St-numbering and a “canonical
4-ordering” known in the area of graph drawings. @ 1997 Elsevier Science B.V.
cf=,
Keywords: Graph partition; Four-connected planar graph; Algorithms; Linear-time algorithms; Canonical decomposition; sr-numbering
1. Introduction
Given a graph G = (YE),
k distinct verE V and k natural numbers
tices UI,UZ,...,Uk
nk such that Et,
ni = IVI, we wish to find
m,n2,...,
Vk of the vertex set V such that
apartitionV1,V2,...,
Ui E L$, 1x1 = ni, and L$induces a connected subgraph
of G for each i, 1 Q i < k. Such a partition is called
a k-partition of G. A 4-partition of a graph G is depicted in Fig. I, where the edges of four connected
subgraphs are drawn by solid lines and the remaining
edges of G are drawn by dotted lines. The problem
of finding a k-partition of a given graph often appears in the load distribution among different power
plants and the fault-tolerant routing of communica* Corresponding author.
’ Email: {nakano,nishi}
@ecei.tohoku.ac.jp.
2 Email: [email protected].
0020-0190/97/$17.00
@
PIf SOO20-0190(97)00083-S
tion networks [ 12,131. The problem is NP-hard in
general [ 11, and hence it is very unlikely that there
is a polynomial-time
algorithm to solve the problem.
Although not every graph has a k-partition, GySri and
Lovkz independently proved that every k-connected
graph has a k-partition for any ut, ~2,. . . , Ilk and
nk [3,8].
However, their proofs do not
nl,n2,...,
yield any polynomial-time
algorithm for actually
finding a k-partition of a k-connected graph. For the
case k = 2 and 3, the following algorithms have been
known:
(i) a linear-time algorithm to find a bipartition of a
biconnected graph [ lo,11 1,
(ii) an algorithm to find a tripartition of a triconnetted graph in O( n2) time, where n is the number of vertices of a graph [ 111, and
(iii) a linear-time algorithm to find a tripartition of a
triconnected planar graph [ 51.
1997 Elsevier Science B.V. All rights reserved.
316
S.-i. Nakano et al. /Information
Processing Letters 62 (1997) 315-322
q=
6
n2= 8
n3 = 7
n4=
10
Fig. 1. A 4-partitioning of a 4-connected plane graph G.
On the other hand, polynomial-time algorithms have
not been known for the case k 2 4. 3
In this paper we give a linear-time algorithm to
find a 4-partition of a 4-connected plane graph G if
~1, ~2, ~3, ~4 are located on the same face of G, as illustrated in Fig. 1. We first bipartition the 4-connected
graph G into two biconnected graphs having about
nl +n;! and n3+n4 vertices respectively, we then bipartitian each of them to two connected graphs, and, by
adjusting the numbers of vertices in the resulting four
graphs, we finally obtain a required 4-partition of G.
To bipartition G into two biconnected graphs, we will
newly define and use a “4-canonical decomposition”
of G, which is a generalization of an St-numbering and
a “canonical 4-ordering” known in the area of graph
drawings [ 2,6,7].
The remainder of the paper is organized as follows.
In Section 2 we introduce our notations and give a
linear-time algorithm to find a 4-canonical decomposition of a 4-connected planar graph. In Section 3 we
present a linear-time algorithm to find a 4-partition of
a 4-connected planar graph. Finally we put our discussions in Section 4.
2. 4-canonical decomposition
In this section we introduce some definitions and
prove that every 4-connected plane graph has a 43A polynomial-time algorithm for any k is claimed in [ 91, but
is not correct [ 41.
canonical decomposition and it can be found in linear
time.
Let G = (YE) be a connected simple graph with
vertex set V and edge set E. Throughout the paper
we denote by it the number of vertices in G, that is,
n = 1VI. An edge joining vertices u and u is denoted
by (u, u). The degree of a vertex u is the number of
neighbors of u in G. The connectivity K(G) of a graph
G is the minimum number of vertices whose removal
results in a disconnected graph or a single-vertex graph
K1. G is called a k-connected graph if K(G) 2 k. We
call a vertex of G a cuc vertex if its removal results
in a disconnected or single-vertex graph. For W C V,
we denote by G - W the graph obtained from G by
deleting all vertices in W and all edges incident to
them.
A graph is planar if it can be embedded in the plane
so that no two edges intersect geometrically except at
a vertex to which they are both incident. A plane graph
is a planar graph with a fixed embedding. The confoour
C(G) of a biconnected plane graph G is the clockwise
(simple) cycle on the outer face. We write C(G) =
wk. WI if the vertices WI, ~2,. . . , wh on
W,W2r...,
C(G) appear in this order. A chord in a biconnected
plane graph G is a path P in G satisfying the following:
(a) P connects two vertices wp and wq, p -C q, on
C(G),
(b) P does not pass through any vertices on C(G)
except the ends wp and wq,
P lies on an inner face, and
ii; there is no edge e on C(G) such that P together
with e forms an inner face.
The chord is said to be minimal if none of w,,+l , wp+2,
. . . , wq_l is an end of a chord. Thus the definition of a minimal chord depends on which vertex
is considered as the starting vertex w1 of C(G).
up_~,up}
be a set of three or more
Let {Ul,UZ,...,
consecutive vertices on C(G) such that the degrees
of the first vertex ~1 and the last one v,, are at
least three and the degrees of all intermediate vertices V2,V3,. . . , up_1 are two. Then we call the set
up--l}
an outer chain of G. Fig. 2 illus{uZ,u3,...,
trates chords, minimal chords and outer chains. The
plane graph G in Fig. 2 has six chords 9, P2,. . . , P6
drawn in thick lines. PI, P2 and P3 are minimal, but
P4, P5 and P6 are not minimal. G has two outer chains
(w5. w6) and {‘+‘9}.
S.-i. Nakano et al./Information Processing Letters 62 (1997) 315-322
Fig. 2. Chords, minimal chords and outer chains.
For a cycle C in a plane graph G, we denote by
I( C, G) the subgraph of G inside C, that is, the plane
subgraph of G induced by the set of vertices and edges
inside (or on) the cycle C. Clearly I( C, G) is biconnected if G is biconnected. We have the following
lemma.
Lemma 1. Assume that G is a 4-connected plane
graphandthatacycleC=Wl,W2,...,wh,wl
inG
is not a face of G. Let wP and wq be the two ends of
any minimal chord in I (C, G) if I (C, G) has a chord,
and let wP = w1 and w4 = wh if I (C, G) has no chord.
Then the .following hold:
(4 ZfW-= {wP+l,wP+2,. . . ,wq-l} isanouterchain
of I (C, G),then I (C, G) - W is biconnected.
(b) Otherwise, there is a set W’ = {wPl, wPf+l,. . . ,
wql} of one or more consecutive vertices on C
such that
(i) p < p’ < q’ < q, and
(ii) none of the vertices in W’ except the jirst
vertex wPf and the last one wql has a neighbor in the proper outside of C.
Moreover, for any of sets W’ satisfying (i) and
(ii), I( C, G) - W’ is biconnected.
Proof. (a) Assume that W = {w,,+l, wp+2,. . . ,
wq_ 1) is an outer chain of Z(C, G).Then Z (C, G) has
a minimal chord with ends wP and wq. Suppose for
a contradiction that I( C, G) - W is not biconnected.
Then I( C, G) - W has a cut vertex u. Since G is
317
4-connected, v must be on C. However, the minimal
chord above passes through v, and v f wP, wq. contradicting to the condition (b) of the definition of a
chord.
(b) Assume that W is not an outer chain of
I( C, G).Then q > p + 2 since G is 4-connected.
Obviously any singleton set W’ = {wP,}, p < p’ < q,
satisfies (i) and (ii). Therefore it suffices to prove
that Z(C, G) - W’ is biconnected for any of sets W’
satisfying (i) and (ii). Suppose for a contradiction
that I( C, G) - W’ is not biconnected for a set W’
satisfying (i) and (ii). Then Z(C, G) - W’ has a cut
vertex v. If v is not on C, then the removal of u and
one or two appropriate vertices in W’ disconnects G
and hence G would not be 4-connected, a contradiction. Thus v must be on C. Assume without loss of
generality that u = wr and q’ < r < q, and that none
of vertices wqj+t, wqt+2,. . . , w,_l is a cut vertex in
I( C, G) - W’. Since v is a cut vertex of Z(C, G) - W’,
Z (C, G) - W’ U {v} is disconnected. Therefore a connected component DI of Z(C, G) - W’U {ZJ) contains
vertices wr+l,w,+2,. . . ,wh,wl,. . . ,wf_l,
and a
connected component D2 of I( C, G) - W’ U {u} contains vertices wqj+t, ~~‘~2,. . . , w,_I. Since Z(C, G)
is biconnected and W (> W’) is not an outer chain
of Z(C,G), an edge el of Z(C,G) joins w, and a
vertex on C(D2) - {w,f+l, wqf+2,. . . , w,-1) and an
edge e2 of I( C, G) joins a vertex ws, p’ < s < r - 1
and a vertex on C(D2) - {w,l+l, wqr+2,. . . , w,._I}.
We choose e2 so that I( C, G) has no edge joining a vertex wst, p’ < s’ < s, and a vertex on
C(D2) - {w,,+l,wq/+2,...,wr-1).
Then Z(C,G)
contains
a chord having ends w, and w, and
passing through el,e2 and some of the edges on
C(D2) - {wqt+lr wqt+2,. . . , w,_I}. Thus the chord
with ends wP and wq would not be minimal, a contradiction.
Cl
Let G = (YE) be a connected graph, and let
(s,t) l E.Wesaythatanordering~=ul,~~,...,u,
of the vertices of G is an St-numbering of G if the
following conditions are satisfied:
(stl)
(st2)
u1 = s and un = t; and
each vi E V - {VI, v,} has two neighbors
and vq such that p < i < q.
Not every connected graph has an St-numbering,
the following lemma holds.
up
but
318
S.-i. Nakano et al. /Information Processing Letters 62 (1997) 315-322
Lemma 2 (see [ 21) . Let G be a biconnected graph,
and let (s, t) be any edge of G. Then G has an stnumbering ?I- = u~,u~,...,v,
such that v1 = sand
ull = t, and v can be found in linear time.
A bipartition of a biconnected graph can be found
by an St-numbering
as follows [ 11,101. Let G =
(YE) be a biconnected
graph, let ut, u2 E V be
two designated distinct vertices, and let nl,n2 be two
natural numbers such that nr + n2 = n. We may assume without loss of generality that (ut , ~42) E E;
otherwise, consider as G the graph obtained from
G by adding a new edge (ur , ~2). Since G is biconnected, by Lemma 2 G has an St-numbering
, un (= ~2). Clearly the following fact
ul(=ul),U2r...
holds:
(st3)
both the subgraphs of G induced by {VI, ~2, . . . ,
vi} and {vi+, , Ui+z, . . . , v,} are connected for
each i, 1 < i < n.
Thus, choosing i = nl, one can find a required bipartition of G in linear time.
Generalizing an St-numbering in a sense, we define
a “4-canonical decomposition” of a 4-connected plane
graph G and in the succeeding section we give an
algorithm to find a 4-partition of G by using the “4canonical decomposition”. We now give the definition
of a 4-canonical decomposition.
Assume
that G = (VIZ) is a 4-connected
plane graph with four designated distinct vertices
ut , up, ug , u4 on the same face of G. We may assume that ur , ~2, ug, u4 lie on the contour C(G)
of G, since, for any face F of G, we can re-embed
G so that F becomes the outer face. We may furthermore assume that the four vertices ut ,u2, ~3, u4
appear on C(G) of G in this order. Moreover we
may assume that (ut,u~), (~3, ~4) E E; otherwise, consider as G the new graph obtained from
G by adding edges (ur,uz)
and (us,uq).
For a
Wi of pairwise disjoint subsets of
set Wl,Wz,...,
V, we denote by Gi the subgraph of G induced by
WI u w2 u . . . U Wi, and by G the subgraph of G
induced by V - WI U W2 U . . . U Wi, that is, G =
G - WI u W2 u . . . U Wi. We say that a partition
Wt of V is a 4-canonical decom17 = Wr,Wz,...,
position of G if the following three conditions are
satisfied:
(co1 )
WI is the set of vertices on the inner face
containing edge (~1, IQ), and WI is the set
of vertices on the inner face containing edge
(~02)
for each i, 1 6 i < 1, both Gi and G are
biconnected, and
for each i, 1 < i < 1, either Wt consists of exactly one vertex on both C (Gi) and C (Gi- 1)
or Wi is an outer chain of Gi or Gi_ 1.
(U3rU4),
(~03)
Fig. 3 illustrates the condition (~03) ; (a) for the
case 1Wil = 1, and (b) and (c) for the cases Wi is
an outer chain of Gi and Gi-t respectively, where Gi
and Gi_r are indicated by different shading and the
vertices in Wt are drawn in black dots.
The 4-canonical
decomposition
defined for 4connected plane graphs is a generalization
of the
“canonical 4-ordering” defined for internally triangulated 4-connected plane graphs [ 6,7].
We have the following two lemmas.
Lemma 3. Let G = (YE)
be a 4-connected
plane graph with four designated distinct vertices
~1, ~42,ug, ~44 appearing on C(G) in this order.
Then G has a 4-canonical decomposition II =
Wt. Furthermore, IT can be found in
Wl,W29...,
linear time.
Proof. Let WI be the set of vertices on the inner face
containing edge (ur , ~2). Clearly Gt is biconnected,
andu3,u4$Wr.Wenowclaimthat~=G-Wris
also biconnected. Let C be the contour of the biconnetted plane graph obtained from G by deleting edge
(ur , ~2). Clearly I( C, G) has neither a chord nor an
outer chain; otherwise, G would not be 4-connected.
Let cycle C start with ~4, then the set WI of vertices are consecutive on C and satisfies (i) and (ii)
in Lemma 1 (b) . Therefore G = I (C, G) - WI is biconnected.
Assume that we have chosen WI, W2, . . . , Wt-1, i >
2, such that the conditions (~02) and (~03) hold for
eachj,l<j<i-l,andthatug,uq$WtUW2U
. . . U Wi- 1. Then we show that there is a set Wi ( c
V-Wt~W2~...uWi-l)suchthat
( 1) Gi is biconnected,
(2) either u3,uq $ Wi Or u3,uq E Wi;
(3) if ~3, u4 $ Wi, then G is biconnected and Wt
satisfies the condition (~03) ; and
S.-i. Nakuno et al. /Infomwtion Processing Letters 62 (1997) 315-322
319
u1
cc>
(b)
(a)
Fig. 3. Illustration of the condition (~03).
(4) ifug,UqEWi,thenI=i,thatis,V=W1UW2U
. . . U Wl, and Wl is the set of vertices on the inner
face containing edge (~3, ~4).
There are the following two cases.
Case 1: Graph Gi_l = G - WI U W2U . . . U Wi-l
is a cycle. In this case Gi-1 is the inner face of G
containing edge (ug , 4). We set 1 = i and Wl = V WI u w2 u *. . U Wi-1. Then u3,u4 E Wl,and V =
WI u w2 u *. . U WI. Since Gi = G, Gi is biconnected.
Cuse2:Othe1~ise.LetC(Gi_1)
=wl,w:!,...,wh,
WI be the contour of Gi-1 with the starting vertex
If Gi_1 has a
WI = ~4. Then Gi_1 = 1(C(Gi-l),G).
chord then let wP and wq be the two ends of a minimal
chord, otherwise let wP = WI = u4 and wq = wr; = ~3.
Let W = {w,+1, wp+2,. . . , wq_l}. We now have the
following three subcases.
Subcase 2a: W is an outer chain of Gi_1. In this
subcase we set Wi = W. Then ~3, u4 $ Wiq and Wi
satisfies (~03). Since G is 4-connected, each vertex in
Wi has at least two neighbors in the biconnected graph
Gi_ 1 induced by WI U W2U . . . U Wi- 1. Therefore the
graph Gi induced by (WI u W2 u ..*U Wi-1) U Wi
is biconnected. By Lemma I (a), G = Gi-l - Wi is
biconnected too.
Subcase 2b: W is not an outer chain of Gi-1, but a
vertex w, in W has two or more neighbors in Gi-1.
In this subcase we set Wi = {w,.}. Then ~3, uq 4 Wi,
and
wr
lies on both C( Gi) and C( Gi-1) and hence
Wi satisfies (~03). Since w, has two or more neighbors in Gi-1, Gi is biconnected. Since Wi = {wr} satisfies (i) and (ii) in Lemma l(b), G = Gi-1 - Wi is
biconnected.
Subcuse 2c: Otherwise. In this subcase, W is not
an outer chain of Gi-1, and every vertex in W has at
most one neighbor in Gi-1. Since G is 4-connected,
W contains two vertices wPt and w,, such that
(1) P < P’ -C 4’ < q,
(2) each of wpt and wqf has exactly one neighbor in
Gi- I and these neighbors are different from each
other, and
(3) none of wpf+l, wpl+2,. . . , ~~1-1 has a neighbor
in Gi-1 a
We now set Wi = {wPl, wpt+l,. . . , ~~8). Clearly
~3, ~4 $ Wi, Gi is biconnected, and Wi is an outer
chain of Gi and hence satisfies (~03). Since Wi satisfies (i) and (ii) in Lemma l(b), G = Gi_1 - Wi is
biconnected.
Thus we have proved that there exists a 4-canonical
decomposition.
One can implement an algorithm for finding a 4canonical decomposition, based on the proof. It maintains a data-structure to keep the outer chains and minimal chords of G. The algorithm traverses every face
at most a constant times, and runs in linear time. Cl
320
S.-i. Nakano et al. /Information Processing Letters 62 (1997) 315-322
Lemma 4. Let WI, W2, . . . , Wl be a 4-canonical decomposition of a 4-connected plane graph G. Then the
following and holdfor any i, 1 < i < 1:
(a) If Wi is an outer chain of Gi as illustrated in
Fig. 3(b), then,forany W/ C Wi, Gi_1 - W: is
biconnected.
(b) If Wi is an outer chain of Gi_1 as illustrated in
Fig. 3(c), then, for any W; G Wi, Gi - W,! is
biconnected.
Proof. We give only a proof for (b) since the proof
for (a) is similar. Let Wi be an outer chain of Gi_t . The
graph Gi-t is biconnected. Since G is 4-connected,
each vertex in Wi has at least two neighbors in Gi-t.
Therefore the graph Gi - W; induced by WI U W2 U
. . . U Wi_1 U (Wi - W;) is also biconnected.
0
3. 4-partition
of 4-connected plane graph
In this section we give our algorithm to find a 4partition of a 4-connected plane graph G. Assume that
the four designated distinct vertices ut , ~2, ug, 244appear on C(G) in this order and nl , n2, ng, n4 are natural numbers such that c$
ni = n.
Algorithm
Four-Partition
Find a 4-canonical decomposition
17 = WI, W2,. . . ,
Wt of G;
Let i be the minimum integer such that ‘$,
1Wjl >
n1 +
n2;,
Letr=~fi=,lWjl-(nt+nz),thatis,ristheexcess
of the number of vertices in WI U W2 U - . - U Wi over
ni +n2;
There are the following
t-2 1;
two cases: ( 1) r = 0, and (2)
Case 1: r =O.
{ In this case, Gi contains nl + n2 vertices, and G
contains n3 + n4 vertices. }
Find a
such
both
Find a
such
both
Return
bipartition &, i$ of the biconnected graph Gi
that ut E VI, u2 E V2, I&l = nl, IV21= n2, and
Vt and V2 induce connected subgraphs;
bipartition Vs, Q of the biconnected graph G
that us E Vs, u4 E V4, IV31= n3, IV’JI= n4, and
& and V4 induce connected subgraphs;
VI, V2, V3, & as a 4-partition of G.
Case 2: r 2 1.
{ In this case, Gi contains nt + n2 + r vertices, and
G=Gi_tWi contains 123+ n4 - r vertices. Since
r 2 1, IWil > 2 and hence Wi is an outer chain of
either Gi_t or Gi. }
L~C(G~-~)=W~,W:!,...,W~,W~
wherewt =uq;
Assume that Wi = {wp+l, wp+2,. . . , wq_l} is an outer
chain of Gi_t as illustrated in Fig. 3 (c) , otherwise,
interchange the roles of ut , u2 and us, u4;
Find an St-numbering ut , ~2.. . . , u,,~+,,_,-~of G such
that s = ut = u4 and t = v~,+,,_~ = 4;
Let wp = up’ and wq = ~1;
Assume that p’ < q’, otherwise, interchange the roles
ofu3 andu4;
There are the following three subcases: (a) n4 < p’,
(b) p’ + r ,< n4, and (c) p’ < n4 < p’ + r;
Subcase 2a: n4 Q pt.
{ In this subcase, the last r vertices in the outer chain
Wi are added to G as the deficient r vertices. }
Let v4 = {UI,U&. . .,a}
be the first n4 vertices in the
st-numbering of Gi;
Let Vi = {um+l, um+2,. . . , u~+,,,_~} be the remaining
123- r vertices in G,
{ By the fact (st3) of an St-numbering both & and &’
induce connected graphs. }
w~_~} be the set of the
Let w; = {wq_t,wq__2,...,
last r vertices in Wi;
Let+quw;;
{ Since wq_t is adjacent to wq in V3),V3 induces a
connected graph of ns vertices. }
Let Gt2 = Gi - W/i
{ Gtz is biconnected by Lemma 2.4(b), and has nl +nz
vertices. }
Find a bipartition VI, V2 of Gt2 such that ut E VI, 142E
V2, IV1I = nl, IV21= n2, and both VI and V2 induce
connected subgraphs;
Return 6, V2, V3, V4 as a 4-partition of G.
Subcase ‘2b: p’ + r < n4.
{ In this subcase, the first r vertices in Wi are added
to G as the deficient r vertices. }
u~_~} be the set of the first n4-r
LetV/={ut,u2,...,
vertices of G, where wp = V,I E Vi;
Let 6 = {u~_~+I,u~_~+~,. . , ,r~~+~,_~} be the remaining n3 vertices of G,
Let W; = {w,+I, wp+2,.
. . , w~+~}
be the set of first r
vertices in Wi;
S.-i. Nakano et al. /Information Processing Letters 62 (1997) 315-322
LetVq=v$Jw;;
{ Vs and & induce connected graphs having n3 and n4
vertices, respectively. }
Let G12 = Gi - W;;
Find a bipartition VI, V2 of the biconnected graph Gt2
such that ut E VI, up E V2, [VII = nl, IV,/ = n2, and
both VI and V2 induce connected subgraphs;
Return 6, V2, Vs, V4 as a 4-partition of G.
Subcase 2c: p’ < n4 < p’ + r.
{ In this subcase, the first n4 -p’ and the last r - (n4 p’) vertices in Wi are added to G as the deficient r
vertices. }
Let W;4 = {w,+l, wP+2,. . . , ~~+,,~-~t} be the set of
the first n4 - p’ vertices in Wi;
Let W& = {w~_I, ~~-2,. . . , w~_-(,._~+,,~~} be the set
of the last r - (n4 - p’) vertices in Wt;
{ Since IW;41 + IW,l,l = r < IWtl, W;4 fl WiJ = 4 and
Iw;4 u w,‘,l = r};
LetV4={UI,Up,...,Up,}UW:4;
Let v3 = (+~+I,
I4 = q,
q++2,.
. . , u~,+~~-~}
U Wi3;
I&( = n3, wP = vP~ E V4, wq E V3,
hence both & and 6 induce connected graphs.
Let Gt2 = Gi - WA U Wi3;
Find a bipartition VI, V2 of the biconnected graph
such that ut E VI, u2 E V2, I&l = nl, IV21= n2,
both VI and VTinduce connected subgraphs;
Return VI, V2, V3, V4 as a 4-partition of G.
{
Clearly the running time of the above algorithm
O(n) . Thus we have the following theorem.
and
}
Gt2
and
is
fact from
4-ordering
Fact 6. For any given internally triangulated
4connected plane graph G = (YE),
two distinct
edges (u,,u2) and (u3,uq) on C(G), and two numbers nl,n2 such that nl + n2 = n and nl ,n2 >
3, there exists a partition V,, V2 of V such that
u1,u2
E
VI,
u3,u4
E
v2,
IhI
=
nl,
and both VI and V2 induce biconnected
ofG.
Iv21 =
Proof. By Lemma 3, G has a 4-canonical
decomposition I7 = WI, W2, . . . , Wt. Since G is internally
triangulated,
each Wt,i = 2,3,. . . , I - 1, is not
an outer chain of Gi or Gi_t and hence each Wi
consists of exactly one vertex on both C( Gi) and
C(Gi_i ). Thus all Wt’s except WI and Wt are singleton sets, each of WI and Wt contains exactly three
vertices, and hence 1 = n - 4. For j, 1 < j < 1,
the vertex in Wj has four or more neighbors, two
of which are in Wt U W2 U +- . U Wj-1 and other
two of which are in Wj+l U Wj+2 U . . . U Wt. Thus,
for j, 1 < j < I, both V, = WI U W2 U . . . U Wj
and V2 = V - (WI U W2 U * * * U Wj) induce biconnected graphs. Hence, it suffices to choose j =
nl - 2. q
4. Conclusion
In this paper we give a linear-time algorithm to
find a 4-partition of a 4-connected plane graph G
in the case four vertices ut , ~2, us, ~44are located on
the same face of G. It is remained as future work
to find efficient algorithms for finding a k-partition
of a k-connected
(not always planar)
graph for
k 3 4.
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