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Decomposition of odd-hole-free graphs by double star cutsets
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Article:
Conforti, M, Cornuejols, G and Vuskovic, K (2004) Decomposition of odd-hole-free graphs
by double star cutsets and 2-joins. Discrete Applied Mathematics, 141 (1-3). 41 - 91 . ISSN
0166-218X
https://doi.org/10.1016/S0166-218X(03)00364-0
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by Double Star Cutsets and 2-Joins
Mihele Conforti
Gerard Cornuejols y
and KristinaVuskovi z
Marh2001, revised November2002
Keywords: Odd-hole-free graphs, even-signable graphs, deomposition, 2-join, double
star utset.
DipartimentodiMatematiaPuraedAppliata,UniversitadiPadova,ViaBelzoni7,35131Padova,Italy. onfortimath.unipd.it
y
GSIA,CarnegieMellonUniversity,ShenleyPark,Pittsburgh,PA15213,USA.g0vandrew.mu.edu z
In this paper we deompose odd-hole-free graphs (graphs that do not ontain as an indued subgraph a hordless yle of odd length greater than three) with double star utsets and 2-joins into bipartite graphs, line graphs of bipartite graphs and the omplementsoflinegraphsofbipartitegraphs.
1 Introdution
In thispaper, all graphsaresimple. A yle is evenif itontains an even numberof nodes,
andit isoddotherwise. A holeisa hordlessylewithat leastfournodes. Anodd-hole-free
graph isa graphthatdoesnotontainan oddhole. When wesay thatagraphG ontains a
graphH,wemean that H appearsin Gasan induedsubgraph.
Given a graph G and a node set S, we denote byGnS the subgraph of G obtained by
removingthe nodes of S and theedges with at least one node inS. A node set S V(G)
is a utset of G if thegraph GnS is disonneted. For S V(G), N(S) denotesthe set of
nodesinV(G)nS that areadjaent to at leastone node inS. A nodeset S isa K m
-starif
S ontainsa liqueC ofsizem and SC[N(C). Wealso refer to aK 1
-starasa star and
to a K 2
-starasa doublestar.
A graphGhasa2-join,denoted byH 1
jH 2
,ifthenodesofGan be partitionedinto sets
H 1
and H 2
with nonempty and disjoint subsets A 1
;B 1
H
1 , A
2 ;B
2
H
2
, suh that all
nodesof A 1
areadjaent to all nodesof A 2
,all nodes of B 1
are adjaent to all nodesof B 2 and these are the only adjaenies between H
1
and H 2
. Also, for i = 1;2, jH i
j> 2 and if
A 1
andB 1
(resp. A 2
andB 2
)arebothofardinality1,thenthegraphinduedbyH 1
(resp.
H 2
) isnota hordlesspath. 2-joinswereintroduedbyCornuejolsand Cunningham[9 ℄.
The mainresult ofthispaperisthe following.
Theorem 1.1 If Gisan odd-hole-free graph, then Gisabipartitegraphor theline graph of
a bipartite graph or the omplement of the linegraph of a bipartite graph, or G has a double
star utset or a 2-join.
In [7 ℄ Conforti, Cornuejols, Kapoor and Vuskovi obtain a polynomial time reognition
algorithmforthelassofeven-hole-freegraphs. Thisalgorithmisbasedonthedeomposition
ofeven-hole-freegraphsby2-joins,doublestarandtriplestar(K 3
-star)utsetsobtainedin[6 ℄.
Itwouldbeofinterestto try touseTheorem1.1 toonstrut apolynomialtimereognition
algorithmforthelassofodd-hole-free graphs. Thisproblemisurrentlynoteven known to
beinNP.
Odd-hole-free graphs are related to perfet graphs introdued by Berge. A graph G is
perfet if every indued subgraph H of G has a hromati number equal to the size of a
largest liquein H. A graphis Berge ifit ontainsneither an oddhole nor its omplement.
Every perfet graph is Berge and the strong perfet graph onjeture (SPGC) states that
every Berge graph is perfet. Wellknown lassesof Berge graphs are bipartitegraphs, line
graphsofbipartitegraphs,andtheomplementsofsuhgraphs. Itiseasytoverifythatthese
graphsareperfet. Ongoingresearh(inMarh 2001)isaimedatobtaininga deomposition
theoremforBergegraphsthatusesmorerenedutsetsthatwouldallowfortheproofofthe
Theorem 1.1. Sine star utsets annot our in minimally imperfet graphs (Chvatal [1 ℄)
and neither an 2-joins (Cornuejols and Cunningham [9 ℄, see also [11 , 4 ℄), it follows that
the strong perfet graph onjeture holds forsquare-free graphs. A skew utset is a utset
S =A[B where A,B aredisjoint andnonempty,and every node ofA is adjaent to every
node of B. Note that a star utset is a skew utset whih itself is a double star utset.
Chvatal [1℄ introdued skew utsets and onjetured that they annot our in a minimally
imperfet graph. This onjeture implies that a deomposition theorem for Berge graphs
similartoTheorem1.1,inwhih\doublestarutsets" arereplaedby\skewutsets",would
prove theSPGC.Suhadeompositiontheoremand theproof oftheskew utsetonjeture
were reentlyobtainedbyChudnovsky,Robertson, Seymourand Thomas [2 ℄.
1.1 Proof Outline
To obtain Theorem 1.1, we prove the following more general result. We sign a graph by
assigning 0,1 weights to its edges in suh a way that, for every triangle in the graph, the
sum ofthe weightsof its edges isodd. A graph Gis even-signableifthere is asigning of its
edges sothat forevery hole inG, thesum of theweightsof its edges is even. Clearly,every
odd-hole-freegraph iseven-signable (assign weight 1to all theedges).
Theorem 1.2 If G is an even-signable graph, then G is a triangle-free graph or the line
graph of a triangle-free graph or the omplement of the line graph of a omplete bipartite
graph, or Ghas a doublestar utset or a 2-join.
The proof outlineof Theorem1.2 isasfollows. Undenedterms willbe denedlater.
Theorem 1.2 holdsfor graphs that ontain no proper wheels and no parahutes (Setion
2).
IfGontainsa properwheelthatisnota beetle, thenGhasadoublestar utset(Setion
3).
If Gontainsan L-parahute, thenGhasa doublestar utset(Setion 4).
If G ontains a T-parahute ora beetle, then G has a doublestar utset orG ontains a
3PC(;)with aType t2,t2p, t4 ort5 node(Setion 6).
If G ontains a 3PC(;) 6= C 6
with a Type t4 or t5 node, then G has a double star
utset(Setion 8).
If G ontains a 3PC(;) with a Type t2 ort2p node, then Ghas a doublestar utset
ora 2-join(Setion 9).
If Gontainsa C 6
withaTypet4 ort5 node,thenGhas adoublestarutset ora 2-join,
Let G be a graph and H an indued subgraphof G. A node v62 V(H) is strongly adjaent
to H,ifjN(v)\V(H)j2.
By C 6
wedenote a holeof length6,and by C 6
its omplement.
A path P is a sequene of distint nodes x 1
;:::;x n
, n 1, suh that x i
x i+1
is an edge,
for all 1 i< n. If n> 1 then nodes x 1
and x n
arethe endnodes of the path. The nodes
of P that are not endnodes are alled intermediate nodes of P. The intermediate nodes of
P arealso referred to asthe interior of P. Where learfrom ontext we write P instead of
V(P). Let x i
and x l
be two nodesof P, wherel i. The pathx i
;x i+1
;:::;x l
is alledthe
x i
x l
-subpathofP andisdenotedbyP xix
l
. AyleCisasequeneofnodesx 1
;x 2
;:::;x n
;x 1
,
n3,suh that thenodes x 1
;x 2
;:::;x n
form a pathand x 1
x n
is an edge. The node set of
a path ora yle Q is denoted byV(Q). The lengthof a path P is the numberof edges in
P and isdenoted byjPj. Similarlythelengthof a yleC is thenumberof edges inC and
isdenoted byjCj.
LetA;B;C bethreedisjointnodesetssuhthatno node ofAisadjaent toanodeofB.
A pathP =x 1
;:::;x n
onnets A to B ifeither n=1 and x 1
hasneighborsin Aand B or
n>1and x 1
isadjaent toat leastone nodeinA andx n
is adjaent to atleastone nodein
B. The path P isa diret onnetion from A to B if, in thesubgraph induedbythe node
set V(P)[A[B,no pathonneting A to B is shorterthan P.
A wheel,denoted by(H;x), isagraphinduedbya holeH andanode x2= V(H) having
at least three neighborsinH, sayx 1
;:::;x n
. Node x isthe enter of thewheel. A subpath
of H onneting x i
and x j
is a setor if it ontains no intermediate node x l
, 1 l n. A
short setor isa setor of length1(i.e. it onsistsof one edge), anda long setor isa setor
of length at least2. A wheel is odd if it ontains an odd number of short setors. A wheel
withk setors isalleda k-wheel.
Alinewheelisa4-wheel(H;v)thatontainsexatlytwotrianglesandthesetwotriangles
haveonlytheentervinommon. Atwinwheelisa3-wheelontainingexatlytwotriangles.
A universal wheel is a wheel (H;v) wherethe enter v is adjaent to all the nodesof H. A
triangle-freewheelisa wheelontainingno triangle. These fourtypesofwheelsaredepited
inFigure1,wheresolidlinesrepresentedgesanddottedlinesrepresentpaths. Aproperwheel
isa wheel thatisnot anyof theabove fourtypes.
A 3PC(x 1
x 2
x 3
;y) is a graph indued by three hordless paths P 1
= x 1
;:::;y, P 2
=
x 2
;:::;y and P 3
=x 3
;:::;y,havingno ommonnodesother than y and suh thatthe only
adjaenies betweennodes of P i
ny and P j
ny, for i;j 2f1;2;3g distint,are the edges of
the lique of size three indued by fx 1
;x 2
;x 3
g. Also, at most one of the paths P 1
;P 2
;P 3
is an edge. We saythat a graphG ontainsa 3PC(;:) ifit ontainsa 3PC(x 1
x 2
x 3
;y) for
some x 1
;x 2
;x 3
;y2V(G).
The followingtheorem isan easy onsequeneof a theoremof Truemper[12 ℄.
Theorem 1.3 ([5 ℄) A graph is even-signableif and only if it does not ontain an oddwheel
or a 3PC(;:).
The fat that even-signable graphs do not ontain odd wheels and 3PC(;:)'s will be
triangle-free wheel
line wheel
twin wheel
universal wheel
z
c
c
c
c
a
a
a
a
d
d
d
d
b
b
b
b
v
v
v
v
z
z
z
Figure2: L-parahutes
2 WP-Free Graphs
In thissetion, we state aresult proven in[3 ℄. First, we needsome denitions.
Denition 2.1 An L-parahute LP(a;db;v;z) is a graph indued by a line wheel (H;v)
where H = a;:::;z:::;b;d;:::;;a, where a, b, , d are the neighbors of v in H, together
with a hordless path P = v;:::;z of length greater than one. Furthermore, no node of
Hnfz;ag is adjaent to an intermediate node of P.
Denition 2.2 A T-parahute TP(t;v;a;b;z) is a graph indued by a twin wheel (H;v)
where H = a;t;b;:::;z;:::;a, where t; a; b are the neighbors of v in H, together with a
hordless path P =v;:::;z oflength greater than one. Furthermore, no node of Hnfz;agis
adjaent toan intermediate node of P.
Denition 2.3 A parahute is eitheran L-parahute or a T-parahute.
Denition 2.4 A graph G isWP-free ifit ontains neither a proper wheelnor a parahute.
Theorem 2.5 LetG bean even-signableWP-freegraphthat isneither atriangle-freegraph
v
t
t
t
z
z
z
a
b
a
b
a
b
v
v
Figure 3: T-parahutes
In fat [3 ℄ proves a stronger result: \double star utset or 2-join" in the statement of
Theorem 2.5 an be replaed by \star utset or universal 2-amalgam". Sine star utsets
annot ourin minimallyimperfet graphs(Chvatal [1 ℄)and universal 2-amalgams annot
our in minimally imperfet Berge graphs (Conforti, Cornuejols, Gasparyan and Vuskovi
[4 ℄), it follows that the strong perfet graph onjeture holds for WP-free graphs. In this
paperwe onlyneedtheweakerstatement 2.5.
As a onsequene of Theorem 2.5, it suÆes to prove Theorem 1.2 when G ontains a
properwheel ora parahute.
3 Proper Wheels
In thissetion, we prove thefollowingtheorem.
Denition 3.1 A beetle isa wheel withfoursetors,exatly two of whih areshort andare
furthermore adjaent.
Theorem 3.2 Let G be an even-signable graph. If G ontains a proper wheel that is not a
beetle, then G has a doublestar utset.
To prove thistheorem, we usea resultof [5 ℄.
AMikeyMouse,denotedbyM(xyz;H 1
;H 2
),isagraphinduedbythenodesetH 1
[H 2 withthe followingproperties:
thenode set fx;y;zg indues alique,
H 1
is a holethat ontains edgexy but doesnotontain node z,
H 2
is a holethat ontains edgexz butdoesnotontain nodey,and
thenode set H 1
[H 2
[5 ℄ Mikey Mousedenedasabove isalled aMikeyMouse withlongears.
A node set S is an extended star if three nodes x;y;z of S indue a triangle and S
N(x)[(N(y)\N(z)). Clearly,an extendedstar utset isalwaysa doublestar utset,sine
SN(x)[N(y).
Theorem 3.3 If an even-signable graph G ontains a Mikey Mouse M(xyz;H 1
;H 2
), then
N(x)[(N(y)\N(z)) is an extended star utset separating nodes of H 1
fromH 2
.
Abutteryisawheel(H;x)withsixsetorsexatlytwoofwhiharelong,and,ifx 1
;:::;x 6 are the neighbors of x in H enountered in thisorder, then x
1 x
2 , x
2 x
3 , x
4 x
5 and x
5 x
6 are
edges. Denote by S 1
and S 2
the two long setors of a buttery (H;x) whose endnodes are
x 1
;x 6
andx 3
;x 4
respetively.
Lemma 3.4 Let G be an even-signablegraph that does not ontain a Mikey Mouseand let
(H;x)bea buttery inG. If u isstrongly adjaent to(H;x)butis notadjaent tox, then u
is oneof the following types.
Type a: All the neighbors of u in H areontained in either S 1
or S 2
.
Type b: The neighbors of u in H are ontained in S 1
[S 2
andu isnot of Typea.
Type : u is adjaent tox 1
;x 2
;x 3
and the neighbors of u in H are all ontained in Hnx 5
,
or u is adjaent tox 4
;x 5
;x 6
and the neighbors of u in H areall ontained in Hnx 2
.
Type d: u is adjaent to x 1
;:::;x 6
andhas possibly more neighbors in S 1
and S 2
.
Type e: u isadjaent to x 2
;x 5
andto no other node of H.
Type f: u has exatly two neighbors in H, that are furthermore adjaent and ontained in
fx 1
;:::;x 6
g.
Proof: Ifuisadjaentto neitherx 2
norx 5
,thenuisofTypeaorb. Sow.l.o.g. assumethat
u is adjaent to x 2
. Suppose that u is adjaent to neither x 1
nor x 3
and is not of Type e.
Thenu musthaveaneighborinS 1
nx 1
orS 2
nx 3
,sayithasaneighborinS 1
nx 1
. Letu 1
be
theneighborofu in S 1
that is losest to x 6
and letS 0 1
bethe u 1
x 6
-subpathof S 1
. If u does
not have a neighborin S 2
, then the node set fu;xg[S 0 1
[S 2
indues a Mikey Mouse. So
u must also have a neighborin S 2
nx 3
. Let u 2
be the neighbor of u inS 2
that is losest to
x 3
,and letS 0 2
bethex 3
u 2
-subpathof S 2
. Then thenode setfu;xg[S 0 1
[S 0 2
induesanodd
wheel withenterx 2
. Soifu is adjaent to neither x 1
nor x 3
,itmustbeof Typee.
We now assume that u is adjaent to exatly one of x 1
;x 3
, say x 1
. Suppose u is not
of Type f. We rst show that u annot have a neighborin S 2
. Suppose it does and let u 1 (resp. u
2
) be theneighborof u inS 2
thatis losest to x 3
(resp. x 4
). If u 1
u 2
is notan edge,
then in the graph indued by S 2
[fu;x;x 2
g there is either a 3PC(x 2
x 3
x;u 1
) (if u 1
= u 2
)
or a 3PC(x 2
x 3
x;u) (if u 1
6=u 2
). If u 1
u 2
is an edge, the node set S 2
[fu;x;x 1
g indues a
3PC(u 1
u 2
u;x). Hene u does not have a neighbor in S 2
. Node u must have a neighborin
S 1
nx 1
,else(H;u)is an odd wheel. Let u 1
be theneighborofu in S 1
nx 1
that islosest to
x 6
. Then theu 1
x 6
-subpathof S 1
togetherwithS 2
;x;x 2
1 3 5 of Type . Assume u is adjaent to x
5
. By symmetry,we an assume that u is adjaent to
bothx 4
;x 6
,and soitis ofTyped. 2
Lemma 3.5 LetGbeaneven-signablegraphthat does notontain MikeyMouses. If(H;x)
is a buttery, then S = N(x)[(N(x 1
)\N(x 3
))nx 2
is a double star utset separating x 2 from the restof H.
Proof: Supposenot andlet P =y 1
;:::;y n
bea diretonnetion from x 2
to Hn(S[x 2
)in
GnS. ByLemma3.4, n>1,y 1
iseithernotstronglyadjaentto HorisofTypeeorf,and
y n
iseithernotstronglyadjaenttoH orisofTypea,bor (adjaent tox 4 ;x 5 ;x 6 andwith
at leastone more neighborin(S 1
[S 2
)nfx 1
;x 3
g).
First we showthat x 4
and x 6
do not have a neighborin P ny n
. Supposenotand let y i be the node of P ny
n
with lowest index that is adjaent to x 4
or x 6
. W.l.o.g. assume y i is adjaent to x
6 . If x
1
does nothave a neighborinfy 1
;:::;y i
g then S 1
[fy 1
;:::;y i
;x;x 2
g
indues an odd wheel with enter x. If x 3
does not have a neighbor in fy 1
;:::;y i
g then
eitherT =S 2
[fy 1
;:::;y i
;x;x 2
;x 6
g induesa Mikey Mouse(if x 4
isnotadjaent to y i
)or
Tnx 6
induesanoddwheelwithenter x(ifx 4
is adjaent toy i
). So x 1
and x 3
bothhave a
neighborinfy 1
;:::;y i
g. Lety j
(resp. y k
)bethenodeof P withlowest indexadjaent to x 3 (resp. x
1
). Then j=1ori, sineotherwiseS 2
[fy 1
;:::;y j
;x;x 2
g induesa Mikey Mouse.
If j = i then fy 1
;:::;y i
;x;x 2
;x 3
;x 6
g indues an odd wheel with enter x 3
. So j = 1 and
hene k 6=1. If k 6=i then S 1
[fy 1
;:::;y k
;x;x 2
g indues a Mikey Mouse. So k = i. But
thenfy 1
;:::;y i
;x;x 1
;x 2
;x 6
g induesan odd wheelwith enter x 1
. Therefore, x 4
and x 6
do
nothave aneighborinP ny n
.
Next we showthatify 1
isnotofTypefthenx 1
andx 3
do nothave aneighborinP ny n
.
Assumeotherwiseandlety i
bethenodeofPny n
withlowestindexadjaentto x 1 orx 3 ,say x 1
. Then S 1
[fy 1
;:::;y i
;x;x 2
g indues a Mikey Mouse. The same argument shows that
if y 1
is of Type f adjaent to x 1
(resp. x 3
) then x 3
(resp. x 1
) does not have a neighborin
P ny n
.
We now onsiderthe followingtwo ases.
Case 1: y n
iseither notstronglyadjaent to H orisof Typea.
W.l.o.g. y n
has a neighborin S 1
. Let u 1
(resp. u 2
) be the neighborof y n
in S 1
that is
losest to x 1
(resp. x 6
). Let S 0 1
(resp. S 00 1
) bethe x 1
u 1
-subpath(resp. u 2
x 6
-subpath)of S 1
.
Node x 3
musthave a neighborinP,sineotherwise S 2
[P [S 00 1
[fx;x 2
g induesa Mikey
Mouse. So y 1
is of Type fadjaent to x 3
,and hene x 1
doesnot have a neighborinP ny n . If u 1 u 2
is not an edge, then P [S 0 1
[S 00 1
[fx;x 2
g indues a 3PC(x 1
x 2
x;:). So u 1
u 2
is an
edge. Lety k
bethenodeofP withhighestindexadjaentto x 3
. ThenS 1
[fy k
;:::;y n
;x;x 3
g
induesa 3PC(u 1 u 2 y n ;x).
Case 2: y n
isof Typeb or.
W.l.o.g. y n
hasa neighborinS 1
nx 6
. Letu 1
be theneighborofy n
inS 1
thatislosest to
x 1
. Let u 2
be theneighborof y n
inS 2
that is losest to x 4
(suh a neighboralways exists).
LetS 0 1 (resp. S 0 2
) bethex 1
u 1
-subpathof S 1
(resp. u 2
x 4
-subpathof S 2
). Ifx 1
doesnothave
a neighborin P ny n
, then P [S 0 1
[S 0 2
[fx;x 2
g indues a 3PC(x 1
x 2
x;y n
). Hene y 1
is of
Type fadjaent to x 1
,and x 3
doesnothave a neighborin P ny n
. Letu 3
(resp. u 4
neighborofy n
in S 2
(resp. S 1
) thatis losest to x 3
(resp. x 6
),and let S 2
(resp. S 1
) bethe
x 3
u 3
-subpathofS 2
(resp. u 4
x 6
-subpathofS 1
). Ifu 3
6=x 4
thenS 00 1
[P[S 00 2
[fx;x 2
gindues
a 3PC(x 2
x 3
x;y n
). OtherwiseP [S 2
[fx;x 2
g induesan odd wheelwith enterx. 2
A bat is omposed of a hordless path y 1
;:::;y n
and a node x suh that, for some 2 <
i<j<n 1,x isadjaent to y k
ifand onlyifk2f1;i;:::;j;ng.
Intheremainderofthissetion,whenwerefertoawheel(H;x)wedenotewithx 1
;:::;x n theneighborsof x inH intheorder in whihthey appear. Fori=1;:::;n, wedenote with
S i
thesetorof (H;x) withendnodesx i
and x i+1
(note x n+1
=x 1
).
Lemma 3.6 Let G be an even-signable graph that does not ontain Mikey Mouses and
butteries. Let (H;x)bea wheel with a bat in G that has fewest number of setors. Suppose
that setors S n
;S 1
;:::;S k
together with node x indue a bat where S n
and S k
are the two
long setors. If node u 2Gn(H [x) is adjaent to x 2
, but not to x and not to x 1
, then u
has no neighbors in Hnfx 2
;x 3
g.
Proof: Sine G ontains no Mikey Mouse, k 3. We rst show that u has no neighbors
in S n
. Suppose not and let u 0
(resp. u 00
) be the neighbor of u in S n
that is losest to x 1 (resp. x
n
). Let S 0 n
(resp. S 00 n
) be the u 0
x 1
-subpath (resp. u 00
x n
-subpath) of S n
. Note that
u 0
6= x n
, sine otherwise S n
[fu;x;x 2
g indues an odd wheel with enter x. Node u must
haveaneighborinHn(S n
[x 2
),else(HnS n
)[S 00 n
[fu;xg induesanoddwheelwithenter
x. If u isadjaent to x i
forsome i2f3;:::;n 1g, then S 0 n
[fu;x;x 2
;x i
g indues an odd
wheel with enter x 2
. Otherwise, there is a shortest subpathS 0
of Hn(S n
[fx 2
;x 3
g) suh
that one endnode of S 0
is adjaent to u and the other to x, and hene S 0 n
[S 0
[fu;x;x 2
g
induesan oddwheel withenter x 2
. Therefore, uhas noneighborsinS n
.
Letx 0 n
betheneighborofx n
inS n 1
andsupposethatuhasaneighborinHnfx 2
;x 3
;x 0 n
g.
ThenthereisashortestsubpathS 0
ofHnfx 2
;x 3
;x 0 n
gsuhthatoneendnodeofS 0
isadjaent
to u and the other to x, and heneS n
[S 0
[fu;x;x 2
g induesa Mikey Mouse. Therefore,
u has no neighbors in Hnfx 2
;x 3
;x 0 n
g. Finally suppose that u is adjaent to x 0 n
. Then u
annotbe adjaent to x 3
,sineotherwise(H;u)isan oddwheel. Node x 0 n
mustbeadjaent
to x,elseS n
[fu;x;x 2
;x 0 n
g induesanoddwheelwithenter x. LetH 0
bethehole indued
by(H nS n
)[u. (H 0
;x) is a linewheel, else the hoie of (H;x) is ontradited. But then
(H;x)isa buttery. 2
Lemma 3.7 IfGisaneven-signablegraphthat hasawheelwitha bat,then thereisadouble
star utset.
Proof: By Theorem 3.3 and Lemma 3.5, we may assume that G ontains no Mikey Mouse
and no buttery. Let (H;x) be a wheel with a batin G thathas fewest numberof setors.
Suppose that setors S n
;S 1
;:::;S k
together with node x indue a bat. Sine G does not
ontain a Mikey Mouse, k 3. Let x 0 1
be the neighbor of x 1
in S n
. We show that S =
(N(x)[N(x 1
))nfx 2
;x 4
;:::;x n
;x 0 1
g is a doublestar utset that separates x 2
from the rest
of H. Supposenotand letP =y 1
;:::;y m
bea diret onnetionfrom x 2
to Hn(S[x 2
)in
GnS. ByLemma3.6, y 1
iseithernotstronglyadjaenttoH orithasexatlytwo neighbors
inH,x 2
and x 3
Supposethaty m
hasaneighborinS n
. Letu (resp. u )betheneighborofy m
inS n
that
is losest to x 1
(resp. x n
). LetS 0 n
(resp. S 00 n
) be theu 0
x 1
-subpath (resp. u 00 x n -subpath) of S n
. If u 0
=u 00
thenP [S n
[fx;x 2
g induesa 3PC(xx 1
x 2
;u 0
). If u 0
u 00
is notan edgethen
P[S 0 n [S 00 n [fx;x 2
ginduesa3PC(xx 1
x 2
;y m
). Sou 0
u 00
isanedge. Supposex 3
hasaneighbor
inP and lety i
beits neighborinP withhighestindex. ThenS n
[fx;x 3
;y i
;:::;y m
gindues
a 3PC(y m
u 0
u 00
;x). So x 3
does nothave a neighborinP. Node y m
must have a neighborin
HnS n
,sineotherwise H[P induesa3PC(y m u 0 u 00 ;x 2
). Henethere isashortestsubpath
S 0
of H nS n
suh that one endnode of S 0
is adjaent to y m
and the other to x. But then
S 0 n
[S 0
[P [fx;x 2
g induesa 3PC(xx 1
x 2
;y m
). Thereforey m
doesnot have a neighborin
S n
.
Node y m
musthave aneighborinHnfx 2
;x 3
g. Let x 0 n
betheneighborofx n in S n 1 . If y m
hasa neighborinHnfx 2 ;x 3 ;x 0 n
g thenthere is a shortestsubpathS 0
of Hnfx 2 ;x 3 ;x 0 n g
suh that one endnode of S 0
is adjaent to y m
and the other to x, and so S n
[S 0
[fx;x 2
g
induesa MikeyMouse. Hene x 0 n
istheuniqueneighborof y m
inHnfx 2
;x 3
g. Node x 0 n
is
adjaent to x,elseS n
[P[fx;x 2
;x 0 n
ginduesanoddwheelwithenterx. Supposex 3
does
not have a neighborin P. Let H 0
be the hole indued by (HnS n
)[P. (H 0
;x) must be a
linewheel,sineotherwiseourhoieof(H;x)isontradited. Butthen(H;x)isabuttery.
Henex 3
hasaneighborinP. Lety i
betheneighborofx 3
inP withhighestindex. Ifi>1,
then S n
[fx;x 2 ;x 3 ;x 0 n ;y i
;:::;y m
g indues an odd wheel with enter x. Hene i = 1. But
thenH[P induesa 3PC(x 2 x 3 y 1 ;x 0 n ). 2
Proof of Theorem 3.2: Assume G has no double star utset. Then by Lemma 3.7, G has
no wheel with a bat. Let (H;x) be a proper wheel that is not a beetle. Assume w.l.o.g.
that S n
is a long setor and S 1
is a short setor. Sine (H;x) is not a wheel with a bat,
either S n
is the onlylong setor, or n >5 and S n
and S n 1
are the only long setors. Let
S =(N(x)[N(x 1
))nfx 2 ;x n ;x 0 1
g,where x 0 1
istheneighborof x 1
in S n
. We laim that S is
a doublestar utset thatseparatesx 2 from S n [S n 1 nfx 1 ;x n 1
g. LetP =y 1
;:::;y m
be a
diretonnetion from x 2 to S n [S n 1 nfx 1 ;x n 1
ginGnS. Letsbetheneighborofx n
in
S n 1
.
Case 1: y m
has aneighborinS n . Let u 1 (resp. u n
) be the neighbor of y m
in S n
that is losest to x 1
(resp. x n
). Let S 0 n (resp. S
00 n
) be theu 1
x 1
-subpath (resp. u n
x n
-subpath) of S n . If u 1 =u n
thenP [S n
[fx;x 2
g induesa 3PC(xx 1
x 2
;u 1
). Ifu 1
u n
isnot anedge, then
P [S 0 n
[S 00 n
[fx;x 2
gindues a3PC(xx 1
x 2
;y m
). Hene u 1
u n
is an edge.
A node of H n(S 1
[S n
) must have a neighbor in P, sine otherwise H[P indues a
3PC(u 1 u n y m ;x 2
). LetubethenodeofHn(S 1
[S n
)thathasaneighborinP and islosest
to x 3
. Let y i
be the node of P with highest index adjaent to u. If u 6= s there exists a
hordless path S 0
from u to x in H n(S 1
[S n
). But then P y i ym [S n [S 0
[x indues a
3PC(u 1 u n y m
;x). Hene u=s.
Suppose that i = m. Sine (H;y m
) is not an odd wheel, u n
= x n
. If s does not have
a neighborin P ny m
, then P [S 0 n 1
[S 0 n
[fx;x 2
g indues a 3PC(xx 1
x 2
;y m
). So s has a
neighborin P ny m
. Let y j
be the neighborof s in P with lowest index. If s6= x n 1 then S n [P y 1 y j
[fx;x 2
;sg indues an odd wheel with enter x. So s = x n 1
then P y
1 y
j [S
n
[fx;x 2
;x n 1
;y m
g indues a 3PC(xx 1
x 2
;x n 1
). So j = m 1. But then
P [fx;x 2
;x n 1
;x n
gindues anoddwheel withenter x n 1
. Thereforei6=m.
Ifi6=1thenP y
i ym
[Hinduesa3PC(u 1
u n
y m
;s). Soi=1. ButthenS n
[P y
i ym
[fx 2
;sg
induesa 3PC(u 1
u n
y m
;y i
).
Case 2: y m
has noneighborsinS n
.
Then S n 1
is a long setor. Let u be the neighbor of y m
in S n 1
that is losest to x n
,
andlet S 0 n 1
betheux n
-subpathof S n 1
. NotethatbythedenitionofS and P,u6=x n 1
.
Then P [S n
[S 0 n 1
[fx;x 2
g induesa 3PC(xx 1
x 2
;x n
). 2
4 L-Parahutes
In thissetion we assume thatGis an even-signable graph. We prove thefollowingresult.
Theorem 4.1 If G ontains an L-parahute, then Ghas a doublestar utset.
Denition 4.2 A rosspath w.r.t. a line wheel (H;x) is a hordless path P =y 1
;:::;y n
in
Gn(H[x)suh that x is not adjaent toany node of P and one of the following holds:
(i) n=1, (H;y 1
) is a line wheel, and eah of the two long setors of (H;x) ontains two
adjaent neighbors of y 1
.
(ii) n> 1, no intermediate node of P has a neighbor in H, y 1
(resp. y n
) has exatly two
neighbors in H that are furthermore adjaent, the neighbors of y 1
in H are ontained
in one long setor of (H;x) and the neighbors of y n
in H are ontained in the other
long setor of (H;x).
Lemma 4.3 If G ontains an L-parahute, then G ontains a line wheelwith no rosspath.
Proof: Suppose Gontains an L-parahute =LP(x 1
x 2
;x 4
x 3
;x;z). LetP be thexz-path
ofnfx 1
;x 2
;x 3
;x 4
g,andletH be thehole induedbyn(Pnz). LetS 1
(resp. S 2
)bethe
longsetor of (H;x)with endnodesx 1
;x 4
(resp. x 2
;x 3
). Supposethat thelinewheel (H;x)
hasa rosspathQ=y 1
;:::;y n
. W.l.o.g. y 1
hasneighborsinS 1
and y n
inS 2
. LetQ 0
bethe
shortest path from y 1
to x in (P [Q[S 2
)nfx 2
;x 3
g. Then S 1
[Q 0
indues a 3PC(;x).
Thereforelinewheel(H;x)has norosspath. 2
Lemma 4.4 If G ontains a line wheel with no rosspath, then G has a doublestar utset.
Proof: Let(H;x)be a linewheel withno rosspath. Let x 1
;x 2
;x 3
;x 4
bethe neighbors of x
in H that appear in thisorder when H is traversed lokwise. W.l.o.g. x 1
x 2
and x 3
x 4
are
edges. LetS 1
(resp. S 2
)bethelongsetors ofH withendnodesx 1
;x 4
(resp. x 2
;x 3
). Letx 0 1 bethe neighborof x
1 inS
1
. LetS =(N(x)[N(x 1
))nfx 0 1
;x 2
;x 3
g. Supposethat S is nota
doublestarutset and letP =y 1
;:::;y n
bea diretonnetion from S 1
to S 2
inGnS. Let
u 1
(resp. u 4
) be theneighborof y 1
in S 1
that is losest to x 1
(resp. x 4
). Let u 2
(resp. u 3
)
betheneighborofy n
inS 2
thatis losest to x 2
(resp. x 3
).
If u 2
= x 3
then (H [P [x)nx 4
ontains a 3PC(x 1
x 2
x;x 3
). So u 2
6= x 3
. Suppose
a node of P n y 1
is adjaent to x 4
and let y i
y i
y n
3 1 2 4 1 4
u 1
= u 4
then (H [P [x)nx 3
ontains a 3PC(x 1
x 2
x;u 1
). So u 1
6= u 4
. If u 1
u 4
is not an
edge, then there is a 3PC(x 1
x 2
x;y 1
). So u 1
u 4
is an edge. If u 2
= u 3
then H[P indues
a 3PC(y 1
u 1
u 4
;u 2
). So u 2
6= u 3
. If u 2
u 3
is not an edge then (H [P [x)nx 4
ontains a
3PC(x 1
x 2
x;y n
). Sou 2
u 3
isanedge. ButtheneitherP isarosspathw.r.t. (H;x),or(H;y 1
)
isan oddwheel. 2
Theorem4.1 follows.
By theresultsof Setions 2-4, itsuÆes to prove Theorem1.2 when Gontainsa beetle
ora T-parahute.
5 Nodes Adjaent to a 3PC(;)
Given node disjoint triangles fa 1
;a 2
;a 3
g and fb 1
;b 2
;b 3
g, a 3PC(a 1
a 2
a 3
;b 1
b 2
b 3
) is a graph
induedbythreehordlesspaths,P 1
=a 1
;:::;b 1
,P 2
=a 2
;:::;b 2
andP 3
=a 3
;:::;b 3
,having
noommonnodesandsuhthattheonlyadjaeniesbetweenthenodesofdistintpathsare
theedges of thetwo triangles. A 3PC(a 1
a 2
a 3
;b 1
b 2
b 3
) is alsoreferred to asa 3PC(;).
Throughout this setion we assume that G is an even-signable graph. By we denote
a 3PC(a 1
a 2
a 3
;b 1
b 2
b 3
) with the three paths P 1
= P a1b1
, P 2
= P a2b2
and P 3
= P a3b3
. For
i=1;2;3, we denote bya 0 i
the neighbor of a i
in P i
and byb 0 i
the neighbor of b i
inP i
. For
distinti;j2f1;2;3g, wedenote byH ij
theholeinduedbyP i
[P j
.
Lemma 5.1 LetGbeaneven-signablegraphandletbea3PC(;). Ifnodeuisadjaent
to ,then it is oneof the following types.
Type tj for j=1;2;3: Node u has exatly j neighbors in and they are all ontained in
fa 1
;a 2
;a 3
g or all in fb 1
;b 2
;b 3
g.
Type p1: Node u has exatly one neighbor in and u is not of Type t1.
Type p2: Node u has exatly two neighbors in , that are furthermore adjaent and are
ontained in P i
, for some i2f1;2;3g.
Type p3: Node u hasat least two nonadjaent neighbors in ,andallthe neighborsof u in
areontained in P i
, for somei2f1;2;3g.
Type p4: Node u has exatly four neighbors in , u 1
, u 2
, u 3
and u 4
, where u 1
u 2
is an
edge that belongs to some P i
, i 2 f1;2;3g, and u 3
u 4
is an edge that belongs to some
P j
, j2 f1;2;3gnfig. Furthermore, u is not adjaent to both a i
and a j
, and it is not
adjaent toboth b i
and b j
.
Type t2p: For distint indies i;j;k 2 f1;2;3g and for z 2 fa;bg, u is adjaent to z i and z
j
, it has at least one neighbor in P k
nfz k
g, and is not adjaent to any node in
(P i
[P j
[fz k
g)nfz i
;z j
g.
Type t3p: Node u has at least four neighbors in . For some z 2 fa;bg, u is adjaent to
z 1
,z 2
and z 3
, and all the other neighbors of u in belong toP i
t6
t1
t2
t3
p1
p2
p3
p4
t2p
t3p
t4d
t4s
t5
1 2 3 1 2 3 1 2 3 1 2 3 fa i ;b j g.
Type t4s: For some i 2 f1;2;3g, N(u) \ fa 1 ;a 2 ;a 3 ;b 1 ;b 2 ;b 3
g = fa 1 ;a 2 ;a 3 ;b 1 ;b 2 ;b 3 g n fa i ;b i
g. Furthermore,ifGdoes notontain aMikeyMouse,thenforj2f1;2;3gnfig,
a j
b j
isnot an edge.
Type t4: Node u is of Typet4d or t4s w.r.t. .
Type tj for j=5;6: Node u is adjaent to j nodes in fa 1 ;a 2 ;a 3 ;b 1 ;b 2 ;b 3
g and possibly
other nodes of .
Proof: Firstweshowthatifforsomei2f1;2;3g,N(u)\fa 1 ;a 2 ;a 3 ;b 1 ;b 2 ;b 3
g=fa 1 ;a 2 ;a 3 ;b 1 ;b 2 ;b 3 gn fa i ;b i
g, then for j 2 f1;2;3gn fig, a j
b j
is not an edge. Suppose not. Assume w.l.o.g.
that i = 3 and a 1
b 1
is an edge. Node u must have a neighbor in P 3
, sine otherwise
P 3 [fa 1 ;a 2 ;b 1
;ug indues an odd wheel with enter a 1
. Let u 3
(resp. v 3
) be the
neigh-borof u in P 3
that is losest to a 3
(resp. b 3
). If u 3
= v 3
then (H 13
;u) is an odd wheel. If
u 3
v 3
is not an edge then P 3 a 3 u 3 [P 3 v 3 b 3 [fa 1 ;b 1
;ug indues a Mikey Mouse. So u 3
v 3
is an
edge, and heneP 3
[fa 2
;b 1
;ug induesa Mikey Mouse.
Assume thatuisnotof Typet1, p1,p2orp3. Then,w.l.o.g. uhasneighborsinbothP 1
and P 2
.
Case 1: u doesnothave aneighborinP 3
.
FirstassumethatuhasauniqueneighborinP 1
orP 2
,sayP 1
. Letu 1
betheneighborof
uinP 1
,andw.l.o.g. assumethatu 1
6=a 1
. Letu 2
betheneighborofuinP 2
thatislosest to
a 2
. Ifu 2
6=b 2
,thenthenodesetP 1 [P 2 a2u2 [P 3
[fuginduesa3PC(a 1 a 2 a 3 ;u 1
). Ifu 2
=b 2
,
theneitheru isofTypet2 orthenode setP 1 a 1 u 1 [P 2 [P 3
[fuginduesa3PC(a 1 a 2 a 3 ;u 2 ).
Now assume that u has at least two neighbors in both P 1
and P 2
. Let u 1
(resp. v 1
) be
the neighborof u in P 1
that is losest to a 1
(resp. b 1
). Let u 2
(resp. v 2
) be the neighbor
of u inP 2
thatis losest to a 2
(resp. b 2
). Firstsupposethat bothu 1 v 1 and u 2 v 2 areedges.
Ifu is adjaent to botha 1
and a 2
,then P 2
[P 3
[fu;a 1
gindues an odd wheel withenter
a 2
. So uisnotadjaentto botha 1
anda 2
,andsimilarlyuisnotadjaent to bothb 1
andb 2
.
Hene u is of Type p4. Now assume w.l.o.g. that u 1
v 1
is not an edge. If u is not adjaent
to all four of the nodes a 1 , a 2 , b 1 and b 2
, then either P 1 a 1 u 1 [P 1 v1b1 [P 2 a 2 u 2 [P 3
[fug or
P 1 a 1 u 1 [P 1 v 1 b 1 [P 2 v 2 b 2 [P 3
[fug induesa 3PC(;u). So u isadjaent to a 1 , a 2 ,b 1 and b 2 ,
and heneit isof Type t4s.
Case 2: u hasa neighborinP 3
.
Fori2f1;2;3g,letu i
(resp. v i
) betheneighborofuinP i
thatislosest toa i
(resp. b i
).
Ifu isadjaent toat mostone node infa 1
;a 2
;a 3
gand at mostone node infb 1 ;b 2 ;b 3 g,then
thenode set P 1 v 1 b 1 [P 2 v 2 b 2 [P 3 v 3 b 3
[fugindues a3PC(b 1
b 2
b 3
;u). So assumew.l.o.g. that u
is adjaent to b 1
and b 2
. If u does nothave a neighbor in (P 1
[P 2
)nfb 1
;b 2
g, then u is of
Typet2p, t3 ort3p. So assumew.l.o.g. that u 1
6=b 1
. Supposeu isnotof Typet4, t5 ort6.
Then u is adjaent to at mostone node of fa 1
;a 2
;a 3
g. If u 2 =b 2 and u 3 =b 3
,then u is of
Typet3p. Otherwise,P 1 a1u1 [P 2 a2u2 [P 3 a3u3
[fuginduesa 3PC(a 1
a 2
a 3
;u). 2
of any ofthepaths of ,and either = C 6
ornoneofthepaths of isan edge.
Type t6b: A nodeu that isof Type t6 w.r.t. , butis notofType t6a.
Lemma 5.2 If u isof Typet6b w.r.t. , then 6= C 6
and u has a neighbor in the interior
of one of the paths of .
Proof: Assume u isof Type t6bw.r.t. ,butu hasno neighborintheinteriorof anyofthe
paths of. Thenw.l.o.g. a 1
b 1
isan edgeand a 2
b 2
is not. Then P 1
[P 2
[uinduesan odd
wheel withenteru. 2
IfnodeuisofTypep3,t2port3pw.r.t. ,thenasubsetofthenodeset[fugindues
a 0
=3PC(;)thatontainsu. Wesaythat 0
isobtainedbysubstitutingu into. Ifu
is ofType t2port3pw.r.t. , and forsome z2fa;bg andi2f1;2;3g, 0
does notontain
z i
,thenwe saythat u isa siblingof z i
.
6 Beetles and T-Parahutes
Theorem 6.1 Let G be an even-signable graph that does not ontain a double star utset.
If G ontains a beetle, then G ontains a 3PC(;) with a Type t2 node. If G ontains a
T-parahute, then G ontains a 3PC(;) witha Type t2,t2p, t4 or t5 node.
Proof: By Theorem3.2, every properwheelof Gis abeetle.
Suppose G ontains a beetle or a T-parahute. For a beetle = (H;v), we denote
the neighbors of v on H by a, t, b and z, where at and bt are edges. For a T-parahute
=TP(t;v;a;b;z), we denote by(H;v) the twin wheel of . In both ases, we denote by
P thepathof from vto z thatuses noedgeof H,andbyH za
and H zb
thesubpathsofH
from z to aand fromz to b thatdo notontain t. LetC bethehole of ontaining b;v;z.
Let S=(N(v)[N(b))nft;m;b 0
g,wherem is theneighborof v inP and b 0
is theneighbor
of binH zb
. Let Q=x 1
;:::;x n
be adiret onnetion fromt to nfa;b;v;tg inGnS.
If x n
has no neighborin C then, sine x n
must have a neighborin H za
na, (na)[Q
ontains a 3PC(bvt;z). So x n
has a neighborin C. If x n
has exatly one neighbor p in C,
thenC[Q[tontainsa3PC(bvt;p). Ifx n
hastwononadjaentneighborsinC,thenC[Q[t
ontainsa3PC(bvt;x n
). So x n
hasexatlytwo neighbors inC and they areadjaent. Then
C[Q[t induesa =3PC(;). By Lemma5.1, ais ofType t2,t2p, t4 ort5 w.r.t. .
When (H;v) is a beetle, bothneighbors ofx n
arein H zb
. It follows from Lemma5.1 thata
isof Typet2. 2
7 Crosspaths and Attahments
Throughout this setion we assume that G is an even-signable graph that ontains a =
7.1 Crosspaths
Denition 7.1 A rosspath w.r.t. = 3PC(;) is a hordless path P = x 1
;:::;x n
in
Gn that satises one of the following:
n=1 andx 1
isof Type p4w.r.t. , or
n>1, x 1
and x n
are of Type p2 w.r.t. , with neighbors in dierent paths of , and
no intermediate nodeof P has a neighbor in .
If x 1
or x n
has neighbors in a path P i
of , we say that P is a P i
-rosspath w.r.t. .
Lemma 7.2 Let=3PC(;)andletP =x 1
;:::;x n
,n>1,bea hordlesspathinGn.
If ; 6=N(x 1
)\ P i
, ; 6= N(x n
)\ P j
, i6= j, and no intermediate node of P has a
neighbor in, then P is a rosspath w.r.t. .
Proof: Suppose that there exist ;P satisfyingthe assumptions of the lemma suh that P
is nota rosspathw.r.t. . Choosesuh ;P withshortest possibleP =x 1
;:::;x n
,n>1.
Assumew.l.o.g. thati=1andj =2. SineN(x 1
)\P 1
andN(x n
)\P 2
,x 1
andx n areofTypet1,p1,p2orp3w.r.t. . SineP isnotarosspathandP
1 ;P
2
aresymmetrial,
we may assume w.l.o.g. that x 1
is of Type t1, p1 or p3 w.r.t. . If x 1
is of Type p3 w.r.t.
, then onsider the 3PC(;) = 0
obtained bysubstituting x 1
into . P nx 1
is not a
rosspathw.r.t. 0
. Therefore, bythe hoie of ;P,thepair 0
;P nx 1
doesnotsatisfythe
assumptions of thelemma. Thusn 1 =1. It follows from Lemma 5.1 that x n
is of Type
p4 w.r.t. 0
, whih is impossibleas x n
has no neighboron P 1
. So we may assume that x 1 is of Typet1 orp1 w.r.t. and similarlythat x
n
is of Type t1, p1 orp2 w.r.t. . Ifx n
is
of Type p2 w.r.t. , thenthe node set P 1
[P 2
[P induesa 3PC(;:). Hene x n
is also
of Type t1 orp1 w.r.t. . Let u 1
(resp. u 2
) be the uniqueneighborof x 1
(resp. x n
) in .
W.l.o.g. u 1
6= a 1
. If u 1
=b 1
and u 2
= b 2
, then the node set P [P 2
[P 3
[fb 1
g indues a
Mikey Mouse. Otherwiseu 2
6=b 2
w.l.o.g. and the node set P 1
[P 2 a 2
u 2
[P 3
[P indues a
3PC(a 1
a 2
a 3
;u 1
Lemma 7.3 Let x be a Type t1 node w.r.t. = 3PC(a 1 a 2 a 3 ;b 1 b 2 b 3
), adjaent to say a 1
.
Suppose that S =(N(a 1
)[(N(a 2
)\N(a 3
)))nx is not a utset and let P =x 1
;:::;x n
be a
diret onnetion from x to nS in GnS. Thenno node of P nx n
is adjaent to a node of
na 0 1
and one of the following holds:
(i) x n
isof Type t1or p1 w.r.t. and its unique neighbor in is in P 1
,
(ii) x n
isof Type p3w.r.t. , withneighborsin P 1
,
(iii) x n
isof Type t2w.r.t. , adjaent tob 2 and b 3 , (iv) x n
isof Type t2p w.r.t. ,adjaent to b 2 andb 3 , (v) x n
isof Type t3p w.r.t. ,adjaent to b 1
, b 2
, b 3
andwith a neighbor in P 1 nb 1 , (vi) x n
isof Type p2w.r.t. , adjaent toa 0 1
, and a 0 1
has a neighbor in P nx n
, or
(vii) x n
is of Type t3 w.r.t. , adjaent to b 1 ;b 2 and b 3 , a 0 1 =b 1 and a 0 1
has a neighbor in
P nx n
.
Proof: Firstweshowthatnonodeof Pnx n
isadjaent toanodeof na 0 1
. Supposenotand
let x i
be thenode of P with lowest index adjaent to a node ofna 0 1
. By the denitionof
S, x i
is adjaent to exatly one of a 2
or a 3
, and no other node of . W.l.o.g. assume x i
is
adjaent to a 2
. Then P x 1 x i [P 2 [P 3
[fx;a 1
g induesa MikeyMouse. Hene,no node of
P nx n
is adjaent to a node ofna 0 1
.
Node x n
annot be of Type t4, t5 and t6 w.r.t. , sine all these types of nodesare in
S. Supposethat x n
is of Type t1 orp1 with theunique neighbor u in. If u is notin P 1
,
thenthe nodeset P 2
[P 3
[P [x indues a 3PC(a 1
a 2
a 3
;u). Similarly,ifx n
isof Typep3,
then it must satisfy (ii), else there is a 3PC(a 1 a 2 a 3 ;x n
). Suppose x n
is of Type p2, with
neighborsuandvin,andw.l.o.g. assumethatuandvarenotinP 3
. Ifa 0 1
hasnoneighbor
in P nx n
, then P 1
[P 2
[P [x induesa 3PC(x n
uv;a 1
). So a 0 1
has a neighborin P nx n
.
Let x i
be thenode of P nx n
with highest indexadjaent to a 0 1
. If x n
is not adjaent to a 0 1 , then P 1 [P 2 [P x i xn
indues a 3PC(x n
uv;a 0 1
). Hene (vi) holds. If x n
is of Type t2 and
it does not satisfy (iii), then w.l.o.g. we may assume that it is adjaent to b 1
and b 3
, and
hene the node set P [P 2
[P 3
[x indues a 3PC(a 1 a 2 a 3 ;b 3
). Supposex n
is of Type t2p
or t3p and does not satisfy (iv) or (v). Then w.l.o.g. x n
is adjaent to b 1
;b 3
and it has a
neighborinP 2
nb 2
,andhene(P[P 2
[P 3
[x)nfb 2
gontainsa 3PC(a 1 a 2 a 3 ;x n ). Suppose x n
isof Type t3. Ifa 0 1
doesnot have a neighborin P nx n
, then P [P 1
[P 3
[x indues a
3PC(x n b 1 b 3 ;a 1
). Soa 0 1
hasaneighborinPnx n
. Supposethata 0 1
6=b 1
andletx i
bethenode
ofPnx n
withhighestindexadjaentto a 0 1
. ThenP x i x n [P 1 [P 3
induesa3PC(x n b 1 b 3 ;a 0 1 ).
Hene (vii) holds. Finallysupposethat x n
is of Type p4 with neighbors inP i
and P j
, for
some i;j2f1;2;3g. Letu i
(resp. v i
) bethe neighborof x n
inP i
that islosest to a i
(resp.
b i
). Similarly deneu j
and v j
. If i=2 and j =3, thenthe node set P 2 a2u2
[P 3 a3u3
[P [x
indues a 3PC(a 1 a 2 a 3 ;x n
). Else we may assume w.l.o.g. that i = 1 and j = 2. Then the
node setP 1 v 1 b 1 [P 2 a 2 u 2 [P 3
x
x
x
x
x
x
x
x
x
x
x
x
1 2 3 1 2 3 1 a
3
. SupposethatS =(N(a 2 )[(N(a 1 )\N(a 3 )))nfx;a 0 2
gisnotautsetandletP =x 1
;:::;x n beadiret onnetion fromx tonS in GnS. Thenno nodeof Pnx
n
isadjaent toanode
of and one of the following holds:
(i) x n
isof Type t1or p1 w.r.t. and its unique neighbor in is in P 2
,
(ii) x n
isof Type p3w.r.t. , withneighborsin P 2
,
(iii) x n
isof Type t2w.r.t. , adjaent tob 1 and b 3 , (iv) x n
isof Type t2p w.r.t. ,adjaent to b 1 andb 3 ,or (v) x n
isof Type t3p w.r.t. ,adjaent to b 1
, b 2
, b 3
, andwith a neighbor in P 2
nb 2
.
Proof: First we show that no node of P nx n
has a neighbor in . Suppose not. By the
denitionof S,the only nodesof that an have a neighborin P nx n are a 1 and a 3 , and
no node ofP isadjaentto botha 1
and a 3
. Supposethatbotha 1
and a 3
haveaneighborin
P nx n
. Then P nx n
ontains a subpathP 0
, suh that one endnode of P 0
isadjaent to a 1
,
theotherto a 3
,andthesearetheonlyadjaeniesbetweenP 0
and. ThenP 0 [P 1 [P 2 [a 3 indues a Mikey Mouse. Now assume w.l.o.g. that onlya
1
has a neighbor in P nx n
. Let
Q be the shortest path from x n
to a 3
in [x n
nfa 1
;a 2
g. Then Q[P [x indues a hole
H and (H;a 1
) is awheel. LetQ 0
bethe shortestpathfrom x n
to a 2
in[x n nfa 1 ;a 3 g. If Q 0
[P [fx;a 3
gindues ahole H 0
,then(H 0
;a 1
)is awheel withone more shortsetorthan
(H;a 1
) and either(H;a 1
) or(H 0
;a 1
) is an odd wheel. Hene H 0
annot bea hole. That is,
either x n
is adjaent to a 3
or theuniqueneighborof x n
in is a 0 3
. If x n
is a Type t1 orp1
nodeadjaentto a 0 3
,thenP 2
[P 3
[P ontainsa 3PC(a 1 a 2 a 3 ;a 0 3
). Ifx n
isof Type p2orp3
adjaent to a 3
, there is a ontradition to Lemma 7.2. If x n
is of Type t2p or t3padjaent
to a 3
, there isa 3PC(b 1 b 2 x n ;a 1
). If x n
is of Type p4 adjaent to a 3
,let u and v beits two
neighborsin P 1
[P 2
. Then P 1
[P 2
[P ontainsa 3PC(uvx n
;a 1
). Therefore, no node of
P nx n
is adjaent to a node of.
Node x n
annot be of Type t4, t5 and t6 w.r.t. , sine all these types of nodesare in
S. Suppose x n
is of Type t1 or p1 with the unique neighbor u in that is in P 1
or P 3
,
say in P 1
. Then the node set P 1
[P 3
[P [x indues a 3PC(xa 1
a 3
;u). Hene if x n
is of
Type t1 or p1, then it must satisfy (i). Similarly, if x n
is of Type p3, then it must satisfy
(ii), else there is a 3PC(xa 1
a 3
;x n
). Suppose that x n
is of Type p2, with neighbors u and
v in . W.l.o.g. assume that u and v are not inP 3
. If x n
is not adjaent to a 1
, then the
node set P 1
[P 2
[P [x indues a 3PC(x n
uv;a 1
). So x n
is adjaent to a 1
. If n=1 then
P 1
[P 3
[fx;x 1
ginduesan oddwheel withentera 1
,and otherwiseP 1
[P 2
[P [fx;a 3
g
induesan odd wheelwithentera 1
. Ifx n
isof Typet2,adjaentto b 2
andsayb 1
,thenthe
nodesetP 1
[P 2
[P[xinduesa3PC(x n b 1 b 2 ;a 1
). Soifx n
isofTypet2,thenitmustsatisfy
(iii). Similarly,ifx n
is of Type t3, then there is a 3PC(x n b 1 b 2 ;a 1
). Ifx n
is of Type t2por
t3p, and itdoesnot satisfy(iv)or(v), thenw.l.o.g. we mayassume that x n
hasa neighbor
inP 3
nb 3
,andhenethenodesetP 1
[P 2
[P[xinduesa3PC(x n b 1 b 2 ;a 1
). Finallyassume
thatx n
isof Type p4withneighborsinP i
andP j
,forsome i;j2f1;2;3g. Letu i
(resp. v i
)
betheneighborofx n
inP i
thatislosestto a i
(resp. b i
). Similarlydeneu j
and v j
. Ifi=1
and j=3,thenw.l.o.g. x n
is notadjaent toa 3
,and hene P 1 v1b1 [P 3 v3b3 [P 2
[P [fx;a 3
1 2 3 n thenode setP
1 v1b1
[P 2 v2b2
[P 3
[P [x induesa 3PC(b 1
b 2
b 3
;x n
). 2
Lemma 7.5 Let x bea Typet3nodew.r.t. =3PC(a 1
a 2
a 3
;b 1
b 2
b 3
), adjaent to saya 1
,a 2 and a
3
. Assume G has no extended star utset, let S =(N(a 2
)[(N(a 1
)\N(a 3
)))nfx;a 0 2
g
and let P = x 1
;:::;x n
be a diret onnetion from x to nS in GnS. Then one of the
following holds:
(i) Nonode of has a neighbor in P nx n
andx n
is of Typep2 or t3w.r.t. .
(ii) n=1, x n
is a sibling of b 1
or b 3
w.r.t. , adjaent toa 1
or a 3
.
(iii) Exatly one of a 1
;a 3
has a neighbor in P nx n
, no other node of has a neighbor in
P nx n
, and x n
is as desribed in Lemma 7.4.
Proof: Firstnote thatbythe denitionof S,nonode ofP an be ofType t4,t5 ort6 w.r.t.
. Also, theonlynodesof that an have aneighborinP nx n
area 1
and a 3
,and there is
no node of P adjaent to both a 1
and a 3
. Supposethat botha 1
and a 3
have a neighborin
Pnx n
. ThenPnx n
ontainsasubpathP 0
suhthatone endnodeofP 0
isadjaent toa 1
,the
otherto a 3
and thesearetheonlyadjaeniesbetweenP nx n
and. ThenP 0
[P 1
[P 2
[a 3 induesa Mikey Mouse. Hene, at most one of a
1 ;a
3
has a neighborin P nx n
. If a 1
ora 3 hasa neighborinP nx
n
thenbyLemma7.4(iii)holds.
Wenowassumethat nonodeof hasaneighborinP nx n
. Supposex n
isof Type t1or
p1andlet ube itsuniqueneighborin. W.l.o.g. assumethatuisinP 1
. Thenthenodeset
P 1
[P 2
[P[xinduesa3PC(xa 1
a 2
;u). Similarly,ifx n
isofTypep3thereisa3PC(;x n
).
Ifx n
isofTypet2,withneighborssayb 1
andb 3
,thenthenodesetP 1
[P 2
[P[fxgindues
a 3PC(xa 1
a 2
;b 1
). If x n
is a sibling of b 2
, there is a 3PC(xa 1
a 2
;x n
). Supposethat x n
is a
siblingof b 1
. Letubetheneighborofx n
inP 1
thatislosest to a 1
. Ifu6=a 1
orn>1,then
P 1 a 1
u [P
2
[P [x indues a 3PC(xa 1
a 2
;x n
). So if x n
is of Type t2por t3pw.r.t. , then
it must satisfy (ii). Finallyassume that x n
is of Type p4 w.r.t. . Then P [ ontains a
3PC(b 1
b 2
b 3
;x n
). 2
Denition 7.6 Fora nodexanda pathP desribedinLemmas7.3,7.4and7.5,wesaythat
the path P isan attahment of node xto . Also, asubset ofthe nodeset[P[x indues
a 3PC(;)that ontains x. Wesaythat this 3PC(;)isobtainedby substitutingx and
its attahment P into .
Theorem 7.7 If G is an even-signable graph that has no extended star utset, then every
node x of Typet1, t2 or t3 w.r.t. =3PC(;) has an attahment P to . Furthermore,
everydiretonnetion fromx tonS (foran appropriateextendedstar S)isanattahment.
Proof: Followsfrom Theorem3.3 andLemmas 7.3, 7.4 and7.5. 2
8 Type t4, t5 and t6 Nodes
Theorem 8.1 Let G bean even-signablegraph that ontains a =3PC(;) and a node
(ii) 6= C 6
and u is of Typet4d w.r.t. , or
(iii) = C 6
, u isof Typet4d w.r.t. ,say adjaent toa 1
;a 2
;b 1
;b 3
,and Gdoesnot ontain
two nodes v and w that are both of Type t4d w.r.t. , uv and uw are not edges, v is
adjaent toa 1
;a 3
;b 2
;b 3
and w isadjaent to a 2
;a 3
;b 1
;b 2
.
Then Ghas a double star utset.
Proof: Suppose G has no doublestar utset. Then by Theorem3.3, G ontains no Mikey
Mouse. LetC be thesetof allorderedpairs;u thatsatisfy(i),(ii)or(iii). Let;u2C. If
u is of Type t4d w.r.t. , then we assume w.l.o.g. thatu is adjaent to a 1
;a 2
;b 1
;b 3
. If u is
of Type t4sw.r.t. , thenwe assume w.l.o.g. thatu isadjaent to a 1
;a 2
;b 1
;b 2
.
Claim 1: If ;u2C satisfy (ii), then G annot ontain nodes v and wthat are of Typet4d
w.r.t. , suh that uv and uw are not edges, v is adjaent to a 1
;a 3
;b 2
;b 3
and w is adjaent
to a 2
;a 3
;b 1
;b 2
.
ProofofClaim1: Supposenot. Thena 1
b 1
mustbeanedge,sineotherwisefa 1
;a 2
;a 3
;b 1
;u;wg
induesanoddwheelwithentera 2
. Alsoa 2
b 2
mustbeanedge,sineotherwisefa 2
;b 1
;b 2
;b 3
;u;wg
indues an odd wheel with enter b 1
. Sine 6= C 6
, a 3
b 3
is not an edge. But then
fa 1
;a 2
;a 3
;b 3
;u;vg induesan odd wheel withenter a 1
. This ompletes theproofof Claim
1.
ByClaim1andthehypothesisinTheorem8.1(iii),wemayassumew.l.o.g. thatif;u2C
and u is of Typet4d w.r.t. , then there is no node v of Type t4d w.r.t. suh thatuv is
notan edgeand v isadjaent to a 1
;a 3
;b 2
;b 3
.
For ;u 2 C dene the orresponding sets S as follows. If u is of Type t4d w.r.t. ,
then let S = (N(u)[N(a 2
))n( nfa 1
;a 2
;b 3
g). If u is of Type t4s w.r.t. , then let
S=(N(u)[N(a 2
))n(nfa 1
;a 2
;b 1
;b 2
g). Sine S isnota doublestarutset, there existsa
diretonnetionP =x 1
;:::;x n
inGnS from (P 1
[P 2
)nS to P 3
nS. LetC 0
beasubsetof
C with the propertythat forall 0
;u 0
2C 0
and all ;u 2C, jN(u 0
)\ 0
jjN(u)\j. Let
;u behosen fromC 0
sothat thesizeof theorrespondingP is minimized.
Claim 2: No node of P isof Type t4, t5or t6 w.r.t. .
Proof of Claim 2: BydenitionofS,nonodeofP isofTypet6w.r.t. . Supposethatsome
x i
is ofType t4 ort5 w.r.t. . Sine x i
annotbe adjaent to a 2
,it mustbe adjaent to a 1 anda
3 . Ifx
i
isadjaent tob 1
,thenfa 1
;a 2
;a 3
;b 1
;u;x i
ginduesanoddwheelwithentera 1
.
So x i
is not adjaent to b 1
, and hene it is of Type t4d w.r.t. , adjaent to b 2
and b 3
. By
the assumption following Claim 1, this annot our if u is of Type t4d w.r.t. . Hene u
is ofType t4sw.r.t. , and so a 1
b 1
is notan edge. But thenfa 1
;b 1
;b 2
;b 3
;u;x i
gindues an
oddwheel withenter b 2
. Thisompletes theproof ofClaim2.
Claim 3: If x i
isof Typep4 w.r.t. , then i=1 and the neighbors of x i
in are ontained
in P 1
[P 2
i i (P
1 [P
2 )nS.
Supposethattheneighborsofx i
inareontainedinP 1
[P 3
. Forj =1;3,letu j
(resp.
v j
)betheneighborofx i
isP j
thatislosesttoa j
(resp. b j
). Firstsupposethatx i
isadjaent
toa 3
. Thenx i
isnotadjaenttoa 1
andso([x i )nP 3 v3b3 induesa 0 =3PC(a 1 a 2 a 3 ;u 1 v 1 x i ). Note that 0 6= C 6
. Sine u is adjaent to a 1
;a 2
;b 1
and it is notadjaent to a 3
;x i
, it must
be of Type t4s w.r.t. 0
. Hene u is adjaent to u 1
and v 1
. Node u annot have neighbors
in P 3
, sine otherwise 0
;u would ontradit the hoie of ;u. So u is of Type t4s w.r.t.
, andhene a 2
b 2
isnotan edge. ButthenP 3
[fa 2
;b 2
;u;u 1
;x i
g induesa3PC(x i u 3 v 3 ;u). Thereforex i
is notadjaent to a 3
.
Let 0
=3PC(a 1 a 2 a 3 ;x i v 3 u 3
) induedby ([x i
)nP 1 v1b1
. Note that 0
6= C 6
. Sine u
is adjaent to a 1
;a 2
and at least one of b 2
;b 3
(i.e. it has aneighborinthe a 2 v 3 -path of 0 ),
and it isnot adjaent to a 3
and x i
,itmustbe ofType t4dw.r.t. 0
. Sine u is adjaent to
b 1
,it hasfewer neighborsin 0
thanin, ontraditing ourhoie of ;u.
Now suppose thatthe neighbors of x i
in areontainedin P 2
[P 3
. By symmetry,the
above proof shows that u is of Type t4d w.r.t. . For j = 2;3 let u j
(resp. v j
) be the
neighborofx i
inP j
thatislosest to a j
(resp. b j
). BythedenitionofS,x i
isnotadjaent
to a 2
, and hene ([x i
)nP 3 v 3 b 3 indues a 0
=3PC(a 1 a 2 a 3 ;v 2 u 2 x i
). Note that 0 6= C 6 .
Sine u isadjaent to a 1
;a 2
;b 1
and itis notadjaent to a 3
;x i
,it must be of Typet4s w.r.t.
0
. Sineu isadjaent to b 3
,uhasfewerneighborsin 0
thanin, ontraditingourhoie
of ;u. Thisompletes theproof ofClaim 3.
Claim 4: If x i
is of Type t2p or t3p w.r.t. , then u is of Type t4d w.r.t. , i=1 and x i is a sibling of b
1 .
Proof of Claim 4: Suppose that x i
is of Type t2p or t3p w.r.t. and let 0
be obtained
from bysubstitutingx i
forits sibling. By thedenitionof S,x i
annot be a siblingofa 1 or a
3
. Supposethat x i
is a siblingof a 2
. Sine u is adjaent to a 1
;b 1
and exatly one node
in fb 2
;b 3
g, and it is not adjaent to a 3
;x i
, it violates Lemma 5.1 w.r.t. 0
. Suppose x i
is
a sibling of b 3
. Sine u is adjaent to a 1
;a 2
;b 1
and it is not adjaent to a 3
and x i
, it must
be of Type t4s w.r.t. 0
. So u is adjaent to b 2
and it is of Type t4s w.r.t. . Sine x i has a neighbor inP
3 nb
3
, i= n. Sine b 1
;b 2
2S, n > 1. But then 0
;u and P 0
= P nx n ontradit our hoie of ;u and P. Supposex
i
is a siblingof b 2
. Then u must be of Type
t4d w.r.t. 0
, and hene w.r.t. too. By thedenition of S, x i
is not adjaent to a 2 , and hene 0 6= C 6
. Alsoi=1 and n>1. But then 0
;u and P 0
=P nx 1
ontradit ourhoie
of ;u and P. Finallysupposethat x i
is a sibling of b 1
. If u is of Type t4s w.r.t. , then
u violates Lemma 5.1 w.r.t. 0
. So u is of Type t4d w.r.t. and t2pw.r.t. 0
. Sine x i
is
adjaent to b 2
,i=1. Thisompletes theproof ofClaim 4.
Claim 5: No node of P isof Type p3w.r.t. .
Proof of Claim 5: Suppose x i
is of Type p3 w.r.t. . Let 0
be obtained from by
substituting x i
into . Note that 6= C 6
. Then 0
;u and P 0
, where P 0
= x 1
;:::;x i 1 or P 0 =x i+1
;:::;x n
,ontradit ourhoieof ;u andP. ThisompletestheproofofClaim5.
Claim 6: n>1, x 1
is eithera sibling of b 1
n
Proof of Claim 6: Followsfrom Claims2,3,4 and 5.
Claim 7: If a 1
has a neighbor in the interior of P, then b 2
and b 3
do not.
Proof of Claim 7: Suppose not. Let x i
and x j
be nodes of the interior of P so that x i
is
adjaent to a 1
,x j
isadjaent to b 2
or b 3
,and no propersubpathof P x
i x
j
hasthisproperty.
By the denition of S, at most one of b 2
;b 3
has a neighbor in the interior of P. Then
P 2
[P 3
[P x
i x
j [a
1
indues a 3PC(a 1
a 2
a 3
;b 2
) or a 3PC(a 1
a 2
a 3
;b 3
). This ompletes the
proof ofClaim 7.
By Claim6, wenow onsiderthe followingases.
Case 1: x n
is ofType t1,p1 orp2 w.r.t. .
Firstweshowthat a 1
doesnothave aneighborintheinterior ofP. Supposenotand let
x i
be the node of P nx n
with highest index adjaent to a 1
. By Claim 7, b 2
and b 3
do not
have a neighbor in the interior of P. If b 1
does nothave a neighborin P xixn
1
, then P xixn ontraditsLemma7.2. Sob
1
hasa neighborinP x
i x
n 1
. By thedenitionofS,u isofType
t4s w.r.t. , and so a 1
b 1
is not an edge. Let x j
be the node of P x
i x
n 1
with highest index
adjaent to b 1
. Then P xjxn
ontraditsLemma 7.2. Therefore a 1
does not have a neighbor
intheinteriorof P.
Next we show that ifb 1
orb 2
has a neighborin the interior of P, then u is of Type t4s
and x n
is of Type t1 w.r.t. adjaent to b 3
. Suppose that b 1
or b 2
has a neighborin the
interior of P. Then, by denition of S, u is of Type t4s. Suppose now that b 3
is not the
uniqueneighborofx n
in. BydenitionofS,b 3
doesnothave aneighborintheinteriorof
P. Let x i
be thenode of P nx n
withhighest indexadjaent to b 1
or b 2
. Ifx i
is adjaent to
exatlyone of b 1
;b 2
,thenP x
i xn
ontraditsLemma 7.2. Hene x i
is adjaent to bothb 1
and
b 2
. Let 0
= 3PC(a 1
a 2
a 3
;b 1
b 2
x i
) ontained in (nb 3
)[P x
i x
n
. Note that u is of Type t4s
w.r.t. 0
. But then 0
;u and P x1xi
1
ontradit ourhoie of;u and P.
Case 1.1: x 1
isof Type t1,p1 orp2 w.r.t. .
First suppose that b 3
is the uniqueneighborof x n
is . Then u is of Type t4s w.r.t.
and so x 1
has a neighbor in (P 1
[P 2
)nfa 1
;a 2
;b 1
;b 2
g. We may assume w.l.o.g. that the
neighborsofx 1
inareontainedinP 2
. Then(nb 2
)[P ontainseithera3PC(a 1
a 2
a 3
;b 3
)
(if b 1
has no neighbors in the interior of P) or a 3PC(a 1
a 2
a 3
;b 1
) (otherwise). So b 3
is not
theuniqueneighborof x n
in, andhene b 1
and b 2
do nothaveneighborsintheinteriorof
P.
If b 3
has a neighbor in theinterior of P, let x i
be thenode of P nx 1
with lowest index
adjaent to b 3
. Then P x
1 x
i
ontradits Lemma 7.2. So b 3
does not have a neighborin the
interior of P. By Lemma7.2 appliedto P,x 1
andx n
mustbothbe of Type p2 w.r.t. .
Suppose that the neighbors of x 1
in are ontained in P 2
. Let u 2
(resp. v 2
) be the
neighborof x 1
in P 2
that is losest to a 2
(resp. b 2
). Let u 3
(resp. v 3
) be the neighbor of
x n
in P 3
that is losest to a 3
(resp. b 3
). Let 0
be the 3PC(u 2
v 2
x 1
;u 3
v 3
x n
) indued by
P 2
[P 3
[P. Supposethat 0
= C 6
. Then a 2
b 2
and a 3
b 3
are edges,and hene u isof Type
t4d w.r.t. . Sine u is adjaent to a 2
and b 3
, and it is notadjaent to P [a 3
, it violates
Lemma 5.1 w.r.t. to 0
. Hene 0
6= C 6
. Let P 0 u 2
u 3
be the u 2
u 3
-path of 0
