Square-free perfect graphs

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Square-free perfect graphs

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White Rose Research Online URL for this paper:

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Article:

Conforti, M, Cornuejols, G and Vuskovic, K (2004) Square-free perfect graphs. Journal of

Combinatorial Theory: Series B, 90 (2). 257 - 307 . ISSN 0095-8956

https://doi.org/10.1016/j.jctb.2003.08.003

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Mihele Conforti

Gerard Cornuejols y

and KristinaVuskovi z

February 2001, revised April2001, August 2002

DipartimentodiMatematiaPuraedAppliata,UniversitadiPadova,ViaBelzoni7,35131Padova,Italy. onfortimath.unipd.it

y

GSIA,CarnegieMellonUniversity,ShenleyPark,Pittsburgh,PA15213,USA.g0vandrew.mu.edu z

ShoolofComputing,UniversityofLeeds,LeedsLS29JT,UK.vuskoviomp.leeds.a.uk

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Weprovethatsquare-freeperfetgraphsarebipartitegraphsorlinegraphsofbipartite graphsorhavea2-joinorastarutset.

1 Introdution

In this paper, all graphs are simple. A hole is a hordless yle with at least four nodes.

A k-hole is a hole with k nodes and a 4-hole is also alled a square. Our mainresult is a

deomposition theorem for graphsthat ontain no odd hole and no 4-hole. (We say that a

graphGontainsH ifH isan induedsubgraphof G). Partialresultsinthisdiretionwere

obtainedbyXue[31℄,[32℄,Fouquet[18 ℄ and reentlybyLinhares-SalesandMaray[21 ℄.

A graph Gis perfet if,inevery induedsubgraph, the sizeof a largest liqueequalsthe

hromati number. A graph G is minimally imperfet ifG is not perfet, but every proper

indued subgraph of G is perfet. The Strong Perfet Graph Conjeture (SPGC), due to

Berge[1℄,statesthatifGisaminimallyimperfetgraph,theneitherGortheomplementof

Gis an odd hole. Thisonjeture was proved reentlyby Chudnovsky,Robertson, Seymour

and Thomas [2℄. In the 70's and 80's, the SPGC was proved for H-free graphs for quite

a number of small graphs H. In fat, the lass of square-free graphs was the last lass of

H-free graphs with jV(H)j = 4 for whih the SPGC was not known: Seinshe [26 ℄ proved

it in 1974 for P 4

-free graphs, Meyniel[22 ℄, [23℄ in1976 forpaw-free graphs(see also Olariu

[24 ℄), Parthasarathy and Ravindra [25 ℄ in 1976 forlaw-free graphs, Tuker [28 ℄ in 1977 for

K 4

-free graphs, Conforti and Rao [13 ℄, [14℄, Fonlupt and Zemirline [17 ℄ and Tuker [29 ℄ in

1987 fordiamond-free graphs(see also[4 ℄).

Given a graphG anda subsetS of its nodes, GnS denotes thesubgraphof Gobtained

by removing the nodes in S and the edges with at least one node in S. A node set S is a

utset of GifthegraphGnS hasmore onneted omponents thanG.

A node set S is a star ifit onsists of a node x and neighborsof x. Chvatal [3℄ showed

thata minimallyimperfet graphannot ontaina star utset.

A graph G has a 2-join, denoted by H 1

jH 2

, with speial sets A 1

;B 1

;A 2

;B 2

that are

nonempty and disjoint, if the nodes of G an be partitioned into sets H 1

and H 2

so that

A 1

;B 1

H 1

,A 2

;B 2

H 2

,all nodesof A 1

are adjaent to all nodes of A 2

,all nodesof B 1 areadjaent to allnodesofB

2

and these aretheonlyadjaeniesbetweenH 1

and H 2

. Also,

fori=1;2, H i

has at leastone pathfrom A i

to B i

andif A i

andB i

arebothof ardinality

1, then the graph indued by H i

is not a hordless path. Cornuejols and Cunningham [16 ℄

showed thataminimallyimperfet graphannot ontain a 2-join.

The line graph of a graph G is the graph H whose nodes are the edges of G and two

nodesr,sofH areadjaent inH ifandonlyiftheedgesr,sofGareinidentto aommon

node of G. It is wellknownand easy to prove that that bipartitegraphs and linegraphsof

bipartitegraphsareperfet.

The mainresult ofthispaperisthe following.

Theorem 1.1 A square-freegraph that ontains no oddhole is bipartiteor the linegraph of

a bipartite graph or has a star utset or a 2-join.

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Proof: Let G be a minimallyimperfet graph. Chvatal [3 ℄ showed that G annot ontain a

star utsetandCornuejolsand Cunningham[16 ℄showedthat Gannotontain a2-join(see

also Cornuejols[15 ℄). Now if G is a square-free graph,by Theorem 1.1, G must be an odd

hole. 2

In fat, we prove thefollowingresult whihis more general thanTheorem 1.1. We sign

a graph by assigning 0,1 weights to its edges. A graph G is even-signable if there exists a

signing suh that, in every triangle the sum of the weights of its edges is odd and in every

holethesum oftheweightsis even. It isobviousthat Gontainsnoodd holesifand onlyif

G iseven-signable and theabove propertyis satisedbyassigningto all theedges of Gthe

weight 1.

Theorem 1.3 Asquare-freeeven-signablegraphistriangle-freeorthelinegraphofa

triangle-freegraph or has a star utset or a 2-join.

Intherstdraftofthispaper(February2001) wemadethefollowingobservations: \The

approah followed in this paper to deompose square-free perfet graphs might be used as

a template for the deomposition of general perfet graphs in the same way as the work

of Conforti and Rao [12℄on linear balanedmatries wasa templatefor the deomposition

of general balaned matries [10 ℄ and [8 ℄, [9 ℄. For general perfet graphs, it is natural to

onsideras basi notonly the bipartite graphsand the line graphsof bipartite graphs, but

alsotheiromplements. Intermsofdeompositions,starutsetsneedtobereplaedbymore

generalutsetssuhasT-utsetsandU-utsets(Hoang[20 ℄)orskewpartitions(Chvatal[3 ℄)."

Interestingly, Chudnovsky, Robertson, Seymour and Thomas [2 ℄ followed suh an approah

to prove theSPGC.

1.1 Even-signable graphs

A wheel, denoted by(H;x), isa graphinduedbya holeH and a nodex2= V(H) havingat

leastthree neighborsinH. Node xistheenterofthewheel. Ifxhask neighborsinH,the

wheelisalledak-wheel. Awheel(H;x)isoddifitontainsanodd numberoftrianglesand

is evenif it ontainsan even numberof triangles: That is,(H;x) is odd (even) ifthe set of

edges ofH havingbothendnodes adjaent to x hasodd(even)ardinality.

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, having no ommonnodesother thany 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 triangle indued by fx 1

;x 2

;x 3

g. Also, at most one of the paths P 1

;P 2

;P 3

has length

1. We say that a graph G ontains a 3PC(;:) if it ontains a 3PC(x 1

x 2

x 3

;y) for some

x 1

;x 2

;x 3

;y2V(G).

It is immediateto hek that ifG ontainsan odd wheelor a 3PC(;:), G isnot

even-signableandhene Gontainsan oddhole. Our proofsusethefollowingharaterization of

even-signable graphs,whihanbederivedfromatheoremofTruemper[27 ℄. (Seealso [11℄).

Theorem 1.4 [6 ℄ A graph is even-signable if and only if it does not ontain an odd wheel

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z

c

a

d

b

v

z

Figure1: L-parahutes

2 WP-Free Graphs

In thissetion, we introduea result proven in[5℄.

Alinewheelisa4-wheel(H;v)thatontainsexatlytwotrianglesandthesetwotriangles

haveonlytheenterv inommon. Note thatifGis alinewheel,thenGisthelinegraphof

atriangle-freegraph. Atwin wheelisa3-wheelontainingexatlytwotriangles. Auniversal

wheelis a wheel (H;v) where the enter v is adjaent to all the nodes of H. A triangle-free

wheelisawheelontainingnotriangle. Aproper wheelisawheelthatisnotanyoftheabove

fourtypes.

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

witha hordless path P =v;:::;z of length greater than one. Nonodeof Hnfz;a;bg andat

most one node among a,b, may be adjaent toan interior 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 of length greater than one. Nonode of Hnfz;a;bg and at most

one nodeamong a, bmay beadjaent to an interior node of P.

AparahuteiseitheranL-parahuteoraT-parahute. AgraphGisWP-freeifitontains

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v

t

z

a

b

a

b

a

b

v

Figure 2: T-parahutes

Theorem 2.3 Let G be a square-free even-signable graph. If G is WP-free, then G is a

triangle-free graph, or the line graph of a triangle-free graph, or G has a star utset or a

2-join.

3 Outline of the proof

Two graphsplayan important role inourproofof Theorem1.3:

A3PC(a 1

a 2

a 3

;b 1

b 2

b 3

)isthegraphontainingnodedisjointtrianglesa 1

;a 2

;a 3

andb 1

;b 2

;b 3

,

plusthree hordless paths, P 1

=a 1

;:::;b 1

, P 2

=a 2

;:::;b 2

and P 3

=a 3

;:::;b 3

,having no

ommon nodes and suh that the only adjaenies between the nodes of distint paths are

theedges of thetwo triangles. A 3PC(a 1

a 2

a 3

;b 1

b 2

b 3

) is alsoreferred to asa 3PC(;).

A beetleis a4-wheel(H;v) ontaining exatlytwo trianglesand these triangleshave two

nodesinommon, theenter v and anode of H.

Nowasa onsequeneofTheorem2.3, intheproofof Theorem1.3, wemayassumethat

Gis asquare-free graph thatontainsa properwheel oraparahute.

-WerstprovethetheoremwhenGontainsaproperwheelthatisnotabeetle(Theorem

3.6),andshowthatifGontainsabeetle,thenGontainsa3PC(;)plusadditionalnodes

adjaent toit (Setion4, Theorem3.4).

-We thenprovethe theoremwhen Gontainsan L-parahute(Setion 5,Theorem4.3).

InalltheresultsthatweobtainfromSetion6on,make useofthefatthatthetheorem

holdswheneverGontainsan L-parahute, oraproperwheelthat isnota beetle.

- In Setion 7 we show that if G ontains a beetleor a T-parahute, then G ontains a

3PC(;) plussome additionalnodesadjaent to it.

- FromSetion 8on,we showthat thetheorem holdswhen Gontainsa 3PC(;).

4 Proper Wheels

Given a subgraph H of a graph G, a node v 62 V(H) is strongly adjaent to H if jN(v)\

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1 n n 1, suh that x

i x

i+1

is an edge, for all 1 i <n. For il,the path x i

;x i+1

;:::;x l

is

alled the x i

x l

-subpath of P and is denoted by P x

i x

l

. The length of a path or a yle is its

numberofedges.

Let A;B;C be three disjoint node sets suh that no node of A is adjaent to a node of

B. A pathP =x 1

;:::;x n

onnets A and B if eithern=1 and x 1

hasneighborsin A and

B,orn>1 and one of thetwo endnodesof P is adjaent to at leastone node inAand the

otherisadjaentto atleastone nodeinB. The pathP isadiret onnetion betweenA and

B if, in the subgraphinduedby the node set V(P)[A[B, no path onneting A and B

is shorter thanP. A diret onnetion P between A and B avoids C ifV(P)\C =;. The

diretonnetion P issaid to be fromA toB ifx 1

isadjaent to some node inAand x n

to

some nodein B.

Lemma 4.1 Let (H;x) be a proper evenwheel and let y bea node that isnot adjaent to x

but has at least 2 neighbors in N(x)\H. Then y has exatly 2 neighborsin H and they are

adjaent.

Proof: Sine G is square-free and y is not adjaent to x, all theneighbors of y inN(x) are

adjaent. Soy has exatlytwoneighborsinN(x)\H,sayy 1

,y 2

andy 1

,y 2

areadjaent. It

remainsto show thaty hasno other neighborinH.

Claim 1: Let Q be a subpath of H whose endnodes are in N(x) at least one of whih is

distint from y 1

, y 2

and no interior node of Q is in N(x). Then Q[fyg ontains an even

number of triangles.

Proof of Claim 1: Assume H ontains a subpath Q with the above properties suh that

Q[fygontainsanoddnumberoftriangles. Thenthisnumbermustbe1 andyhasexatly

two neighbors in Q, else Q[fx;yg indues an odd wheel with enter y. Let x i

, x j

be the

endnodes of Qand y i

,y j

be the adjaent neighbors of y inQ, losest to x i

,x j

respetively.

SineGissquare-free andthepairy i

,y j

isdistintfrom y 1

,y 2

,we mayassumew.l.o.g. that

x i

and y i

are distint.

Let x 0 i

,x 0 j

betheneighborsofx i

,x j

inHnQ. Sine(H;x)isa propereven wheel,xhas

a neighborinHnfx 0 i

;x i

;x j

;x 0 j

g. So ify also hasaneighborinHn(Q[fx 0 i

;x 0 j

g),wehave a

3PC(yy i

y j

;x). ThereforeN(y)\HQ[fx 0 i

;x 0 j

g. Sine(H;y) has apositive even number

of triangles,y i

, y j

arethe onlytwo neighborsof y inQand x i

6=y i

,it follows thatx j

=y j

.

So bothx andy areadjaentto x 0 j

. LetH 0

be theholex i

;x;x 0 j

;y;y i

;Q y

i x

i

. Then (H 0

;x j

) is

an odd wheel. Thisonludes Claim1.

Sine y is adjaent to y 1

, y 2

,it follows byClaim 1 that ify hasat leastthree neighbors

inH,then(H;y)ontains anodd numberof trianglesand is anodd wheel. 2

Let (H;x)beawheel thatontainsat leastonetriangle. Asetorof(H;x)isa maximal

subpath Q of H suh that Q[fxg is a triangle-free graph. Two setors Q i

and Q j

are

adjaentif Q i

[Q j

[fxgontainsat least one triangle. Notethat a setormay have length

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sets R andB (thered andbluenodes) sothatnodesinthesame setorhavethe sameolor

whilenodes inadjaent setors have distintolors.

Lemma 4.2 Let (H;x) be a proper even wheel that is biolored and let y be a node that is

not adjaent to x but has at least two neighbors in H painted with distint olors. Then y

has exatly 2 neighbors in H and these nodesare endnodesof adjaent setors of (H;x).

Proof: Let y 1

, y 2

be neighbors of y in H havingdistint olors, suh that one of the y 1

y 2

-subpathsofH,sayQ,ontainsnootherneighborofy. Siney 1

,y 2

havedistintolors,then

Q[fxg ontains an odd number of triangles. This number must be 1 and x has exatly

two neighbors in Q, elseQ[fx;yg indues an odd wheel with enter x. Let x 1

, x 2

be the

neighborsofx inQ, losest to y 1

,y 2

respetively.

If x and y have two ommonneighborsinH, we aredonebyLemma 4.1. So we assume

w.l.o.g. that y 2

6=x 2

. Lety 0 1

,y 0 2

betheneighborsof y 1

,y 2

in HnQ.

Claim 1: Nodex has a neighbor in Hn(Q[fy 0 1

;y 0 2

g).

Proof of Claim 1: Assume not. Sine (H;x)is a propereven wheel,theny 1

=x 1

,node x is

adjaent to both y 0 1

, y 0 2

and y 0 1 y

0 2

is not an edge. Node y has at least three neighbors in H,

elsewe have an odd wheel withenter x, buty is notadjaent to y 0 1

or y 0 2

elsewe are done

byLemma4.1. Lety 3

betheneighborofy inHnQlosest toy 1

andlet y 00 2

betheneighbor

of y 0 2

inH,distintform y 2

.

Ify 3

6=y 00 2

,there isanoddwheel(H 0

;y 1

)whereH 0

=y;y 2

;y 0 2

;x;H y

0 1 y

3

;y. So y 3

=y 00 2

and

we have a3PC(y 0 1

y 1

x;y 00 2

). Thisompletes theproof ofClaim 1.

If y has a neighbor in Hn(Q[fy 0 1

;y 0 2

g), by Claim 1, we have a 3PC(x 1

x 2

x;y). So all

the neighbors of y in H are inluded infy 0 1

;y 1

;y 2

;y 0 2

g. Now y 1

= x 1

and both x and y are

adjaent to y 0 1

, else we have an odd wheel with enter x. But now we are done by Lemma

4.1. 2

Denition 4.3 A diamond is a graph with four nodesand ve edges. Conneted diamonds

D(a 1

a 2

a 3

a 4

;b 1

b 2

b 3

b 4

) onsist of two node disjoint sets fa 1

;:::;a 4

g and fb 1

;:::;b 4

g eah of

whih indues a diamond suh that a 1

a 4

and b 1

b 4

are not edges, together with four paths

P 1

;:::;P 4

suh that for i=1;:::;4, P i

is an a i

b i

-path. Paths P 1

;:::;P 4

arenode disjoint

and the only adjaenies betweenthem are the edges of the two diamonds.

Lemma 4.4 Let R , B be a bioloring of a proper evenwheel (H;x) and let P =y 1

;:::;y n

,

n>1, be a hordless path with the following properties:

1) y 1

is adjaent toa node in B and tono node in R , y n

is adjaent to a node in R and

no node in B, and y 1

, y n

have nonadjaent neighbors in H.

2)No node of P is adjaent tox and no interior node of P is adjaent toa node of H.

Then (H;x)is a beetle and (H;x)[P indues onneted diamonds.

Proof: Lett 1

,t 2

beblue andredneighborsofy 1

,y n

inH suhthatoneof thet 1

t 2

-subpaths,

sayQ 12

ofH isshortest. Then Q 12

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12 1 2 neighbors,losestto t

1 ,t

2 inQ

12

respetively. Sine(H;x)isaproperwheeland y 1

,y n

have

nonadjaentneighborsinH,theny 1

ory n

hasatleastone otherneighborinH,elsethereis

an odd wheelwith enterx.

Assumew.l.o.g. thaty 1

hasatleastoneotherneighborandlett 3

besuhnode,losest to

t 2

inHnQ 12

. Lett 4

betheneighborofy n

,losesttot 3

inthet 3

t 2

-subpathofHnt 1

. (Possibly

t 4

=t 2

). LetQ 34

be the t 3

t 4

-subpathof Hnt 1

. By theabove argument, x has exatly two

neighborsinQ 34

. Letx 3

,x 4

bethese two neighbors,losest to t 3

,t 4

inQ 34

respetively. We

usethe notationH =t 1

;Q 12

;t 2

;Q 24

;t 4

;Q 34

;t 3

;Q 13

;t 1

. If t 2

and t 4

oinide,Q 24

has length

0.

Noweithert 1

andt 3

areadjaentort 2

=t 4

=x 4

,elsewehave a3PC(x 1

x 2

x;y 1

). Assume

theseond ase doesnothold: So t 1

and t 3

are adjaent. Sine t 1

t 3

are bothblue nodes, x

annot beadjaent to both. So, sine(H;x) isnot alinewheel,x hasat leastone neighbor

x

inHnfx 1

;x 2

;x 3

;x 4

gandbyonstrution x

isaninteriornodeofQ 24

. Thisimpliesthat

t 2

andt 4

aredistintand nonadjaent and inH[P there isa 3PC(y 1

t 1

t 3

;y n

).

So theseond ase musthold, i.e. t 2

=t 4

=x 4

. Furthermore sinex 1

,x 2

aretheunique

neighborsofx inQ 12

,we musthave x 2

=t 2

=t 4

=x 4

.

Sine (H;x) is not a twin wheel, x has at least one other neighbor, say x

, and by

onstrutionx

isan interiornodeofQ 13

. Nowx

mustbeadjaent tobotht 1

,t 3

,elsethere

is a 3PC(x 1

x 2

x;y 1

) or a 3PC(x 3

x 4

x;y 1

). Finallyt 1

6=x 1

and t 3

6=x 3

, else (H;x) is either

an odd wheel oran universal wheel. So (H;x)is a beetle. Finallyx

is adjaent to y 1

,else

we have a3PC(x 1

x 2

x;t 1

). Therefore (H;x)[P indues onneteddiamonds. 2

Theorem 4.5 LetG bea square-free even-signablegraph that ontains a proper evenwheel

(H;x). Furthermore if(H;x)isa beetle,assume that no onneted diamonds ontain (H;x).

Let R ;B be a bioloring of the nodes in H and assume w.l.o.g. that B nN(x) is nonempty.

Then (N(x)[x)nR isa star utset of G, separating R from BnN(x).

Proof: Assume not and let P = y 1

;:::;y n

be a diret onnetion from R to B nN(x) in

Gn(N(x)[x[H). Let y k

be the node of lowest index with a neighbor in B that is not

adjaent to all theneighbors ofy 1

inR . (Possibly k=n). By Lemma 4.2, k>1. So y k

has

no neighborinR . If nonode of P withindexsmallerthan k is adjaent to a node inB,the

theoremholdsasaonsequeneofLemma4.4. Letr2R beadjaentto y 1

andlety j

,j<k,

be thenode of lowest indexadjaent to a node b2B. Sine j <k, band r areadjaent. If

j = 1,by Lemma 4.2, b and r are the uniqueneighbors of y 1

in H and if j > 1,sine G is

square-free and b is adjaent to all the neighborsof y 1

inH, r is theunique neighbor of y 1 inH. Letz betheneighborofr inHnfbg.

Sine (H;x) isa propereven wheel and BnN(x) is nonempty, x hasa neighborb 1

2B

thatisadjaenttoneitherbnorz. Letb 2

beaneighborofy k

inHnfz;r;bg,sothattheb 1

b 2

-pathT inHnfz;r;bgisshortestandletQ=y k

;b 2

;T;b 1

;x. ThenH 0

=x;r;y 1

;P y1y

k ;y

k ;Q;x

isa hole.

Claim 1: If z2B, then z has no neighbor in H 0

nfx;rg.

Proof ofClaim1: Assumenot. Thenboth(H 0

;b)and(H 0

;z)arewheels. If(H 0

;b)isaproper

wheel,thelaim followsbyLemma4.2appliedto z anda bioloringof (H 0

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if(H ;z) isa properwheel, thenb ontradits Lemma4.2. So (H ;b) and (H ;z) are either

linewheelsortwinwheels. There arethree possibilitiesdependingon whether bothare line

wheels,botharetwinwheelsoroneisalinewheelandtheotherisatwinwheel. Sineband

z are notadjaent to b 1

, itan be veried thatin all three ases there is an odd wheel ora

squareand thisprovesClaim1.

Letb 0

betherstneighborofy k

,enounteredwhentraversingHnfrgfromztobandletS

betherb 0

-subpathofHnfbg. ByClaim1,zhasnoneighborinP y1y

k

,sor;y 1

;P y1y

k ;y

k ;b

0 ;S;r

isa hole. Sine b 0

2B and r 2R , S[fxgontainsanodd numberof triangles. Thereforex

is adjaent to r,z and to no other node of S, elsethere inan odd wheel with enter x. In

partiular,byClaim1,b 0

6=z, sob 0

62N(x)and y k

=y n

.

Let x 0

be the rst neighborof x, enountered when traversing Hnfrg from b 0

to b and

let T be the b 0

x 0

-subpath of H nfbg. If b 0

is the unique neighbor of y k

in T, there is a

3PC(xrz;b 0

). Ify k

hasexatlytwoneighborsinT,sayb 0

,b 00

,andb 0

,b 00

areadjaent,thereis

a 3PC(y k

b 0

b 00

;x). Finallyify k

hastwo nonadjaent neighborsinT,there isa3PC(xrz;y k

).

2

The followingresult isan immediateonsequeneof theabove theorem.

Theorem 4.6 LetG bea square-freeeven-signablegraph. If G ontains a proper wheelthat

is nota beetle, then G has a star utset.

5 L-Parahutes

In this setion, we assume that G is a square-free even-signable graph that ontains an

L-parahute=LP(a;db;v;z).

We use the followingnotation. The two triangles are av and bdv. The top path is P d (thed-pathofHnb) andweindiatewith

0 ,d

0

theneighborsofandd inP d

. Thebottom

path isP ab

(theab-pathofHnd). ThemiddlepathisP mz

,wherezinaninteriornodeinP ab and m isa neighborof v inP.

Lemma 5.1 Letybeanodenotadjaenttovbutadjaenttoatleasttwonodesinfa;b;;dg.

Then N(y)\ is eitherfa;g or fb;dg.

Proof: SineGissquare-free,N(y)\fa;b;;dgiseitherfa;gorfb;dgandweassumew.l.o.g.

thatN(y)\fa;b;;dg=fa;g.

Claim 1: P d

[fyg ontains an evennumber of triangles.

Proof of Claim 1: Assume not. Ify ontainsaneighborinP d

nf; 0

g,siney isadjaent to

,P d

[fv;yg induesanoddwheelwithentery. So, 0

aretheonlyneighborsofy inP d and P

d

[fv;a;ygindues anodd wheelwithenter . ThisprovesClaim1.

TheargumentusedinClaim1alsoshowsthatP ab

[yontainsanevennumberoftriangles.

So a;are theonlyneighborsof y in H=P d

[P ab

,else(H;y) isan odd wheel. Suppose y

hasa neighborinP mz

. Note that y is notadjaent to m sineotherwisev;;y;m indues a

square. LetR be a shortestpath fromy to b iny[P mz

[P bz

nm. Then (R[P d

[y;v) is

(11)

d in P

ab [P

mz

. Then N(y)\ is either fa;g or fb;dg.

Proof: ByLemma5.1,itsuÆestoassumethatyhasat mostoneneighborinfa;b;;dgand

derivea ontradition.

Claim 1: Nodey has exatly two neighbors in P d

and they areadjaent.

Proof of Claim 1: Assume not, then y has either 1 neighboror 2 nonadjaent neighbors in

P d

.

If y 1

is the unique neighbor of y in P d

, assume w.l.o.g. that y 1

6= . When y has a

neighborinP ab

[P mz

nfb;mg there is a 3PC(av;y 1

). Sine G is square-free, y annot be

adjaent to both b and m. If y is adjaent to b then it is not adjaent to d and there is a

3PC(av;b). Som istheuniqueneighborofy inP ab

[P mz

. Ify 1

=dthenv;m;d;y indues

a square. So y 1

6=d. W.l.o.g. a does not have a neighbor in P mz

nz. But then there is a

3PC(av;m).

Assume y has nonadjaent neighbors in P d

. Sine y has at most one neighbor in

fa;b;;dg, there is a 3PC(av;y) or a 3PC(bdv;y) whenever y has a neighbor in P ab

[

P mz

nfmg. So m is theunique neighborof y in P ab

[P mz

and assume w.l.o.g. that a does

nothave aneighborinP mz

nz. Then there isa 3PC(av;m) and thisompletesClaim1.

SoP d

[fygontainsauniquetriangle,sayyy 1

y 2

. IfyhasaneighborinP ab

[P mz

nfa;bg,

there isa 3PC(yy 1

y 2

;v). So y isadjaent to aorbandno other nodeof P ab

. Sine yhas at

mostone neighborinfa;b;;dg, P d

[P ab

[y induesan oddwheel withenter y. 2

Theorem 5.3 Let G be a square-free even-signable graph. If G ontains an L-parahute,

then G has a star utset.

Proof: Weshowthat,ifGontainsan L-parahute=LP(a;db;v;z) thenS=v[N(v)n

fa;b;mg is astar utset ofG, separatingP d

nf;dg fromP ab

[P zm

.

Assume not and letP =y 1

;:::;y n

be a diretonnetion from P d

nf;dg to P ab

[P mz in GnS. We assume w.l.o.g. that and P are hosen so that the ardinality of the node

setof [P is minimized. ByLemma5.2n>1. Italso followsfromourassumptionthat at

mostone of ,d has a neighborin P nfy n

g and if is adjaent to a node inP nfy n

g,then

0

and possibly are theonlyneighbors ofy 1

in . From nowon, we assume that d has no

neighborinP nfy n

g.

Claim 1: Nonode of P nfy n

g is adjaent to .

Proof of Claim 1: Assume not. Then 0

and possibly arethe onlyneighborsof y 1

in.

Case 1: Node y 1

istheonlyneighborof inP nfy n

g.

If y n

has a neighbor in P ab

[P zm

nfa;bg, by Lemma 5.2, y n

is not adjaent to or d

and there is a 3PC(y 1

0

;v). If y n

is adjaent to a, possibly and to no other node of

there is an odd wheel with enter . If y n

is adjaent to b, d and to no other node of

there is a 3PC(y 1

0

;d). Finally, ify n

is adjaent to b and to no other node of there is a

3PC(y 1

0

(12)

1 n If y

n

hasa neighborin P ab

[P zm

nfm;bg,byLemma 5.2,y n

isnotadjaent to d. Then

P ab

[P zm

[fy n

gnfm;bgontainsay n

a-pathQ 1

andP ab

[P zm

[fy n

gontainsay n

b-pathQ 2 suh that H

1

= d;P d

0 ;

0 ;y

1 ;P;y

n ;Q

1

;a;v;d and H 2

= d;P d

0 ;

0 ;y

1 ;P;y

n ;Q

2

;b;d are both

holes. Nowone of(H 1

;), (H 2

;)isan odd wheel. Ify n

isadjaent to bandhasnoneighbor

inP ab

[P zm

nfm;bgthereisa3PC(av;b). Finallyifmistheonlyneighborofy n

inP ab

[P zm there isa 3PC(av;m) ora 3PC(bdv;m). ThisompletesClaim 1.

By the minimalityof [P, y 1

has either a uniqueneighboror two adjaent neighbors

inP d

.

Assume that y 1

has a uniqueneighbor, sayy

,in P d

. If y n

is adjaent to ord, say d,

byLemma5.2, y n

isadjaent tob,dandto noothernodeofand thereisa3PC(y n

bd;y

).

If y n

hasa neighborinP ab

[P zm

nfmgthere is a 3PC(av;y

) ora 3PC(bdv;y

). So m is

theuniqueneighborofy n

in and there isa 3PC(av;m) ora3PC(bdv;m).

So y 1

has two neighbors, say y 0

, y 00

in P d

and y 0

, y 00

are adjaent. By Claim 1, y 1

is

not adjaent to ord. If y n

has a neighborin P ab

[P zm

nfa;bg, by Lemma 5.2, y n

is not

adjaent to ord and hene there isa 3PC(y 0

y 00

y 1

;v). ByLemma 5.2y n

is notadjaent to

bothaand b. So w.l.o.g. a isthe uniqueneighborof y n

inP ab

[P zm

. By Lemma 5.2, y n

is

notadjaent to d. If y n

isnotadjaent to there is a3PC(y 0

y 00

y 1

;a)and otherwisethere is

a 3PC(y 0

y 00

y 1

;). 2

6 Nodes Adjaent to a 3PC(;)

We denote by a 3PC(a 1

a 2

a 3

;b 1

b 2

b 3

) with the three paths P 1

= P a

1 b

1 , P

2 = P

a 2

b 2

and

P 3

=P a3b3

. For i=1;2;3, we denote bya 0 i

the neighbor ofa i

inP i

and byb 0 i

theneighbor

of b i

inP i

. Fordistint i;j 2f1;2;3g, we denotebyH ij

thehole induedbyP i

[P j

.

Lemma 6.1 LetGbeaneven-signablegraphandletbea3PC(;). Ifnodeuisadjaent

to ,then it is oneof the following types.

Type tj for j=1;2;3: Nodeuhasexatlyjneighborsinandtheyareeitherallontained

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 , whih are furthermore adjaent and

on-tained 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

(13)

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 to P i

for some i2f1;2;3g.

Type tj for j=4;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

othernodesof. Furthermore,ifuisofTypet4,thenuhastwoneighborsinfa 1

;a 2

;a 3

g

and two in fb 1

;b 2

;b 3

g.

Proof: Assume that u is notof Type p2 or p3. Then, w.l.o.g. u has neighbors in both P 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 a 2

u 2

[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 a1u1

[P 1 v 1

b 1

[P 2 a2u2

[P 3

[fug or

P 1 a 1

u 1

[P 1 v1b1

[P 2 v2b2

[P 3

[fug induesa 3PC(;u). So u isadjaent to a 1

,a 2

,b 1

and b 2

,

and heneit isof Type t4.

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 ot t6.

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

Nodes adjaent to are furtherlassiedasfollows.

Type t6a: AnodeuthatisofTypet6w.r.t. , suhthatnoneofthepathsofisanedge

and uhas noneighborsintheinteriorof anyofthe pathsof .

Type t6b: A nodeu that isof Type t6 w.r.t. butisnotof Typet6aw.r.t. .

Type t4d: Fordistint i;j2f1;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 j

(14)

t6

t1

t2

t3

p1

p2

p3

p4

t2p

t3p

t4d

t4s

t5

(15)

1 2 3 1 2 3 1 2 3 1 2 3 i i

Lemma 6.2 LetGbeasquare-freeeven-signablegraphwithnostarutset. If=3PC(a 1

a 2

a 3

;b 1

b 2

b 3

)

then the following holds.

(i) If u is of Typet6b w.r.t. , then u isadjaent to all nodes of .

(ii) If u is of Type t5 w.r.t. , say not adjaent to a 3

, then none of the paths of is an

edge and N(u)\=fa 1

;a 2

;b 1

;b 2

;b 3

;b 0 3

g.

(iii) If u is of Type t4d w.r.t. , say not adjaent to a 3

and b 2

, and a 1

b 1

is not an edge,

then N(u)\=fa 1

;a 2

;a 0 2

;b 1

;b 3

;b 0 3

g, fa 1

;a 2

;b 1

;b 3

;b 0 1

g or fa 1

;a 2

;a 0 1

;b 1

;b 3

g.

(iv) If u isof Typet4s w.r.t. , say not adjaent to a 3

and b 3

, then a 1

b 1

and a 2

b 2

are not

edges.

Proof: Let u be of Type t6 w.r.t. . Fori;j 2f1;2;3g, (H ij

;u) must be a linewheel or a

universalwheel. If(H 12

;u)isalinewheel,thensois(H 13

;u),and henenoneofthepathsof

isanedgeand uhasnoneighborsintheinteriorof anyof thepaths of,i.e. u isofType

t6aw.r.t. . If (H 12

;u) isa universal wheel,then sois (H 13

;u), and hene u is adjaent to

all nodesof.

Letu be ofTypet5w.r.t. ,saynotadjaent toa 3

. Supposethat (H 12

;u)isauniversal

wheel. Sine G is square-free, a 1

b 1

and a 2

b 2

annot bothbe edges. W.l.o.g. a 1

b 1

is not an

edge. Butthen(H 13

;u)isaproperwheelthatisnotabeetle,ontraditingTheorem4.6. So

(H 12

;u)annot be a universalwheel, and hene it must be a linewheel. But then (H 13

;u)

must be a beetle, and so(ii)holds.

Let u be of Type t4d w.r.t. , say not adjaent to a 3

and b 2

. Suppose a 1

b 1

is not an

edge. (H 12

;u) must be a beetle or a line wheel. Suppose (H 12

;u) is a line wheel. Then

N(u)\(P 1

[P 2

) =fa 1

;a 2

;b 1

;b 0 1

g. Ifu has a neighborin P 3

nb 3

,then(H 13

;u) isa proper

wheelthatisnotabeetle,ontraditingTheorem4.6. Sonowassumethat(H 12

;u)isabeetle.

Supposethatuisadjaenttoa 0 1

. ThenN(u)\(P 1

[P 2

)=fa 1

;a 2

;a 0 1

;b 1

g. Ifuhasaneighbor

on P 3

nb 3

, then (H 13

;u) is a properwheel that is not a beetle, ontraditing Theorem 4.6.

Finallysupposethatu isadjaent to a 0 2

. Then N(u)\(P 1

[P 2

)=fa 1

;a 2

;a 0 2

;b 1

g. Butthen

(H 23

;u)mustbe alinewheel and heneN(u)\P 3

=fb 3

;b 0 3

g.

Let u be of Type t4s w.r.t. , say not adjaent to a 3

and b 3

. Supposea 1

b 1

is an edge.

SineGissquare-free,a 2

b 2

isnotanedge. NodeumusthaveaneighborinP 3

,sineotherwise

P 3

[fa 1

;a 2

;b 1

;ug indues an odd wheel with enter a 1

. So (H 13

;u) must be a line wheel.

Butthen (H 23

;u)is aproperwheel thatis notabeetle,ontraditing Theorem4.6. Soa 1

b 1 isnot anedge, and similarlyneitheris a

2 b

2

. 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 and i2f1;2;3g, 0

does notontain

z i

,thenwe saythat u isa siblingof z i

(16)

To prove themainresult of thissetion,we needthe followinglemma,whose proof appears

in[5 ℄.

Lemma 7.1 LetGbeaT-parahute TP(t;v;a;b;z)that isnotan L-parahuteandsuhthat

no proper subgraph of G is a parahute or a proper wheel. Then G is one of the following

graphs,see Figure2.

Type a: Nointerior node of P isadjaent to aor b.

Type b: An interior node of P is adjaent to a or b, say a, (C b

;a) is a triangle-freewheel

and ais adjaent toz.

Type : An interior node of P is adjaent toa or b, say a,(C b

;a) isa twin wheel and ais

adjaent toz.

Theorem 7.2 LetGbeasquare-freeeven-signablegraph. AssumethatGhasno3PC(;)

witha Typet2,t2port4node. ThenGisatriangle-freegraph,orthe linegraphofa

triangle-freegraph, or G has a star utset or a 2-join.

Proof: By Theorem 2.3, the result holds for WP-free graphs. By Theorem 4.6, the result

holds when G ontains a proper wheel that is not a beetle. By Theorem 5.3, the result

holds when G ontains an L-parahute. So we may assume that G ontains a beetle or a

T-parahute.

LetbeabeetleoraT-parahuteTP(t;v;a;b;z). IfisaT-parahute, byLemma7.2,

weassumethatisofTypea,borofFigure2. Ifisabeetle(H;v),denotetheneighbors

ofv ontheholebya;t;b;zwhereatvandtbvarethetriangles. IfisaT-parahute, denote

by (H;v) its twin wheel with enter v. We assume w.l.o.g. that if is a T-parahute of

Type, thenG ontainsno beetleand no T-parahuteof Type a orb. We denote byP the

pathoffrom v toz thatuses noedgeof H andbyH za

andH zb

thesubpathsofH from z

to aand fromzto bthatdo notontain nodet. W.l.o.g. bhasnoneighborintheinteriorof

P. LetC betheholeofontainingb;v;z. Considerthestar S=(v[N(v))nft;mgwhere

m is the neighbor of v in distint from a;b;t. Assume that S is not a utset separating

t from B = V()nfa;b;v;tg and let Q= x 1

;:::;x n

be a diret onnetion from t to B in

GnS. No node ofQ isadjaentto bothaand b sineGissquare-free.

Case 1: n=1,orn>1 and no nodeof Q x

1 x

n 1

is adjaent to aorb.

Node x n

has at least one neighbor in C if is a T-parahute of Type b or and, by

symmetry,we assume w.l.o.g. that thisisalso the asewhen isa beetleora T-parahute

of Type a.

Ifx n

hasexatlyone neighborp inC,thenthereis a3PC(bvt;p)sinep isdistintfrom

band v.

Assume x n

has exatly two adjaent neighbors in C. Assume rst that one of these

neighborsis b and theother isb 0

adjaent to b in H zb

. Ifn=1, there is an odd wheel with

enter b. So n>1. If x n

has no neighborinH za

nz,there is a 3PC(x n

bb 0

;t). Let z 0

(17)

neighborof z in H za

. If x n

has a neighbor in H za

nfz;zg, then there is a 3PC(x n

bb;v).

Thereforex n

isadjaent to z 0

. Ifb 0

6=z,(H;x n

) isan oddwheel. So b 0

=z and (H;x n

) is a

twinwheel. NowQ[nbinduesa 3PC(;)andbis aTypet4snodewithrespetto it,

sotheresult holds.

Nowassumethatbothneighborsofx n

inCaredistintfromb. Thenthereisa3PC(;)

withanodeofTypet2(whenisabeetleorofTypea)orofTypet2p(whenisofTypeb)

orof Type t4d(whenis ofType ). Thereforethe resultholds.

Assume x n

hastwo nonadjaent neighbors in C. Then there is a 3PC(bvt;x n

) ifn >1

orif x n

is notadjaent to b. So assume n=1 and x 1

is adjaent to b. If x 1

hasa neighbor

inH za

nz,there is an odd wheelwith enter t. So all theneighborsof x 1

in areinC[t.

If is a beetle, there is a 3PC(avt;z). So is a T-parahute. Assume rst that is of

Typea. If x 1

has noneighborinP,there is a3PC(avt;z). If x 1

hasauniqueneighborpin

P,there is a 3PC(avt;p). Therefore x 1

has several neighbors inP. Node z is adjaent to b

sine, otherwise, there is an odd wheel withenter t. Let pbe the neighborof x 1

in P that

is losest to z. Then H za

[P zp

[ft;x 1

g indues a hole H 1

and, ifp 6= z, (H 1

;b) is an odd

wheel. Sop =z. Let z 0

bethe neighborof z inP. Ifx 1

has a neighborinP nfz;z 0

g, there

is a 3PC(bzx 1

;v). So the neighbors of x 1

in P are exatly z and z 0

and therefore (C ;x 1

) is

a twinwheel. Then (nb)[x 1

induesa 3PC(;)and bis aType t4dstronglyadjaent

node withrespetto it, so theresult holds. Now assume thatis a T-parahute of Type b

or . Node x 1

is not adjaent to m sine, otherwise, m;v;t;x 1

wouldbe a square. Sine x 1 hasa neighborinC distintfrom band itsneighborinC,there isan oddwheel withenter

t.

Case 2: n>1and some node ofQ x

1 x

n 1

isadjaent to aorb.

Let x j

besuha nodewith lowest index.

Assume rst that j2. Denote byd the node among a;b that is adjaent to x j

. Let R

be a shortestpath from x n

to v in nfa;b;tg and let R 0

be a shortestpath innft;v;dg

from x n

to the node d 0

among a;b that is distint from d. Let H 1

be the hole indued by

R[Q[t and H 2

the hole indued by R 0

[Q[t. We willshow that the wheel (H 1

;d) is

a properwheel that is not a beetle. (H 1

;d) is nota universalwheel sine d is not adjaent

to x 1

nor a triangle-free wheel sine d is adjaent to v and t, nor a twin wheel sine x j

is

adjaent to d but not to t or v. Suppose (H 1

;d) is a line wheel. Then (H 2

;d) is a proper

wheel. Suppose(H 2

;d)isabeetle. Thenj=n 1andzisadjaenttobothx n

andd,andx n hasa neighborinP

vz

nz. Notethat ifd =b,thenaz annot bean edge(sinebz is anedge

andH annotbeasquare) andheneannotbeaT-parahuteofTypeaorb. Ifx n

hasa

neighborinP nfz;z 0

g(wherez 0

istheneighborofz inP),thenthere isa3PC(tvd 0

;x n

). So

zand z 0

aretheonlyneighborsofx n

inP,and henethereisa3PC(4;4)withaTypet4d

node and the result holds. Finally,suppose(H 1

;d) is a beetle. Then d =a, is of Type

and x n

is adjaent to m. If x n

has no other neighbor in , there is a 3PC(tvd 0

;m). If x n hasaneighbordistintfrommand zinC,thereisa3PC(tvd

0 ;x

n ). Ifx

n

hasexatlym and

z as neighbors, there is a 3PC(;) with node d being of Type t4d relative to it. So the

result holds. Therefore (H 1

;d)is a properwheel thatis nota beetle. So theresult holdsby

Theorem4.6.

Assume nowthat j=1. Denote byd thenode amonga;b thatis adjaent to x 1

.

Case 2.1: Nonode of Q x

2 xn

(18)

n 1 and let H

3

be the hole R 1

[Q. Then (H 3

;v) is an odd wheel unless d = b and is a

T-parahute of Type . Butthen Q[nv induesa 3PC(;) and v is astrongly adjaent

node ofTypet4d withrespetto it. Sotheresult holds.

Nowassume somehordlesspathR 2

from x n

to tinnfd;vgdoesnotontainm. IfR 2 doesnot ontain neighbors of d, then let H

4

be thehole (R 2

nt)[Q[fv;dg. Then (H 4

;t)

is an odd wheel. So R 2

ontainsa neighborof d. LetH 5

be the hole R 2

[Q. Then (H 5

;d)

isan oddwheel.

Case 2.2: Somenode ofQ x2xn

isadjaent to aorb.

Letx k

besuhanodewithlowestindex. Ifx k

is notadjaentto d,thenQ x1x

k

[fa;v;bg

indues a hole H 6

and (H 6

;t) is an odd wheel. So x k

is adjaent to d. Let H 7

be a hole in

Q[nfa;bg that ontains v. Sine d is adjaent to v;t;x 1

and x k

, the wheel (H 7

;d) is a

properwheel ora universalwheel.

Case 2.2.1: (H 7

;d) isa properwheel.

If (H 7

;d)is notabeetle, theresultholdsbyTheorem4.6. Soassume (H 7

;d)is abeetle.

Ifd=athenannotbea T-parahuteofType. Thenodeinfa;bgnfdghasnoneighbor

in Q, sine otherwise a subpath of Q together with a, t and b indues an odd wheel with

enter d. Let R be a hordless path from x n

to t in nfd;vg. Some interior node of R

must be adjaent to d sine, otherwise, there is an odd wheel with enter d. x n

must have

a neighborin distint from theneighbord 0

of d inH zd

sine, ifd 0

were theonlyneighbor

of x n

in, theassumption that(H 7

;d) isa beetlewouldbe ontradited. Thisimpliesthat

d 0

=zandthatx n

hasallits neighborsinP. Soisnotabeetle. LetH 8

betheholeR[Q.

(H 8

;d) is a wheel. Sine d is adjaent to t and x 1

but not its neighbors on H 8

, it is not a

beetle. So if (H 8

;d) is a proper wheel,the result holdsbyTheorem 4.6. If (H 8

;d) is not a

properwheel,it mustbe alinewheel. This impliesthatk =nand that x n

is adjaent to z.

Ifd=b then,sinebz isanedge, az annotbe anedge(elseH isa square) andsoannot

be a T-parahute of Type . annot be a T-parahute of Type b sine, otherwise, d =a

and there is a 3PC(azx n

;v). So isa T-parahuteof Type a. Sine (H 7

;d) is a beetle, x n hasaneighborinPnz. Ifx

n

hasaneighborinPnzdistintfromtheneighborz 0

ofz,there

is a3PC(dzx n

;v). So x n

hasexatly three neighbors in, namely d;z and z 0

. Inthisase,

Q[ndinduesa 3PC(;)and thenode d isastronglyadjaent node ofTypet4dwith

respetto it. Sothe resultholds.

Case 2.2.2: (H 7

;d) isa universalwheel.

Then is a T-parahuteof Type and d=a. LetH 9

bethe holeinQ[nfa;vg that

ontains b. Then (H 9

;a) is a proper wheel that is not a beetle unless n = 2 and x n

has a

neighborinCnfz;m;v;bg. Sinen=2,x 2

isnotadjaenttom(otherwisethereisa5-wheel

withentera)butx 2

isadjaenttozsine(H 7

;a)isauniversalwheel. Letz 0

betheneighbor

ofz onC distintfrom mandp theneighborofx 2

losestto binHnfa;zg. Supposep6=z 0

and let H 10

be the hole induedby H bp

[fv;m;z;x 2

g. Then (H 10

;a) is an odd wheel. So

p =z 0

. But now na[fx 1

;x 2

g indues a 3PC(;) and node ais a Type t4s node. So

(19)

8 Dierent Paths

In thissetion, we assumethat Gisan even-signable graph.

Denition 8.1 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 interior node of P has a neighbor in . See Figure4

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. .

Denition 8.2 A weakonnetion w.r.t. =3PC(a 1

a 2

a 3

;b 1

b 2

b 3

) is a hordless path P =

x 1

;:::;x n

in Gn suh that no interior nodeof P has a neighbor in ,N(x 1

)\=p and

N(x n

)\=q, p6=q,andeither both p, q are ontained in fa 1

;a 2

;a 3

gor bothareontained

in fb 1

;b 2

;b 3

g.

Theorem 8.3 Let G bean even-signablegraph, let =3PC(;)and let P =x 1

;:::;x n

,

n>1,bea hordless path in Gn. If ;6=N(x 1

)\P i

, ;6=N(x n

)\P j

,i6=j, and

no interior nodeof P has a neighbor in ,then P isa rosspath or aweakonnetion w.r.t.

.

If, furthermore, Gis square-free and has no star utset, then P is always a rosspath.

Proof: Assume w.l.o.g. that i=1 and j =2. Sine N(x 1

)\P 1

and N(x n

)\ P 2

,

x 1

and x n

are of Type t1, p1,p2 orp3 w.r.t. . Suppose thatP is not a rosspath. Then

w.l.o.g. x 1

is of Type t1, p1 or p3 w.r.t. . If x 1

is of Type p3 w.r.t. , then we onsider

the3PC(;) obtainedbysubstitutingx 1

into . It follows from Lemma6.1that x n

is of

Type t1,p1, p2 or p3 w.r.t. thisnew 3PC(;). So we mayassume that x 1

(20)

n n w.r.t. , then the node set P

1 [P

2

[P indues a 3PC(;:). Hene x n

is also of Type t1

or p1 w.r.t. . Let u 1

(resp. u 2

) be the unique neighbor of 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 P is a weak onnetion, and otherwise the node set

P 1

[P 2 a 2

u 2

[P 3

[P indues a3PC(a 1

a 2

a 3

;u 1

).

NowassumethatGissquare-freeandhasnostarutset. Suppose=3PC(a 1

a 2

a 3

;b 1

b 2

b 3

)

hasa weak onnetion P =x 1

;:::;x n

from a 1

to a 2

. Let S =(N(a 1

)[a 1

)nfx 1

;a 0 1

g. Sine

S is not a star utset, there exists a diret onnetion Q = y 1

;:::;y m

from P to nS in

GnS. ThenodesofP[P 1

[P 3

[a 2

indueaMikeyMouseasdenedin[6 ℄. BytheMikey

Mousetheorem [6 ℄,some node ofQ isadjaentto botha 2

and a 3

. ChooseP;Q suh thatQ

isasshortaspossiblesubjettotheonditionthatP isaweakonnetion withone endnode

adjaent to a 1

and theother adjaent to eithera 2

or a 3

. Let y k

be thenodeof lowestindex

adjaent to botha 2

and a 3

.

Claim 1: If k 2, no node y 1

;:::;y k 1

is adjaent toa 2

or a 3

.

Proof of Claim 1: Suppose notand let y j

be the node of lowest index adjaent to a 2

ora 3 (note that j = 1 is possible). Node y

1

has a unique neighbor on P and this neighbor is

x n

sine, otherwise, our hoie of P;Qwould be violated. If y j

isadjaent to a 3

,there is a

3PC(a 1

a 2

a 3

;x n

). So y j

is adjaent to a 2

. Let y i

, j < i k, be the node of lowest index

adjaent to a 3

. Then W =(Q y

1 y

i a

3 a

1 P;a

2

) is a wheel that isnot triangle-free, universal, a

twin wheel or a beetle. Suppose it is a line wheel with triangles a 2

x n

y 1

and a 1

a 2

a 3

. Then

i< k and therefore there is an L-parahute with middlepath a 2

;y k

;y k 1

;:::;y i

. But then

G has a star utset byTheorem 5.3, a ontradition. So W is a properwheel that is not a

beetle. But then G has a star utset byTheorem 4.6, a ontradition. This ompletes the

proof ofClaim 1.

Let S 0

=(N(a 2

)[a 2

)nfx n

;a 0 2

g. Sine Ghasno star utset,there is adiret onnetion

Q 0

= z 1

;:::;z p

from P to nS 0

in GnS 0

. By the Mikey Mouse theorem applied to the

Mikey Mouse induedby thenodesP [P 2

[P 3

[a 1

,there is nodeof Q 0

adjaent to both

a 1

anda 3

. Letz l

bethenodeoflowestindexinQ 0

adjaent tobotha 1

anda 3

. Letz t

bethe

node oflowestindexinQ 0

that isadjaent to a 1

ora 3

.

Case 1: t<l

Supposerst that z t

is adjaent to a 3

. Then z 1

is adjaent to a 1

;P in a triangle sine,

otherwise,there is a 3PC(a 1

a 2

a 3

;:). Letu;v bethe neighborsof z 1

on P. Ifu;v6=a 1

,then

there is a 3PC(z 1

uv;a 1

). So u or v oinides with a 1

. But then there is an L-parahute

induedbya subpathofP [Q 0 z1z

l [fa

1 ;a

2 ;a

3

g. So, byTheorem5.3, there is a star utset,

a ontradition.

So z t

isadjaent toa 1

. Letz r

bethenodeofQ 0

withlowestindexadjaenttoa 3

. Clearly

t < r l. Let H be the hole passing through a 3

in Q 0

[P [a 3

. Then (H;a 1

) is a wheel

that is not triangle-free, universal or a twin wheel. If r < l, then the wheel (H;a 1

) is not

a beetle. Suppose it is a line wheel with triangles a 1

x 1

z t

and a 1

a 2

a 3

. Then there is an

L-parahutewithmiddlepathbeinga subpathofa 1

;z l

;z l 1

;:::;z t

. If(H;a 1

)isa linewheel

with triangles a 1

z t

z t+1

and a 1

a 2

a 3

, then there is an L-parahute with middle path being a

subpathof a 1

;z l

;:::;z r

(21)

xn

x1

xn

Figure5: Pathsfrom a Type t1 node

W is a proper wheel that is not a beetle. But then G has a star utset by Theorem 4.6, a

ontradition. Thereforer=l. Now (H;a 1

) isnotan L-wheel. Sine Gdoesnothave a star

utset,(H;a 1

)annotbeaproperwheelbyTheorem4.6,soitmustbeabeetle. NotethatQ

isa pathfromthetopofthebeetleto thebottom thatdoesnotindueonneteddiamonds

withthe nodesof thebeetle(sine y k

isadjaent to a 2

), ontraditingLemma 4.4.

Case 2: t=l

Let R be a shortest path from y k

to z l

in P [Q[Q 0

. If R ontains neither x 1

nor x n

,

then R[fa 1

;a 2

;a 3

g indues an odd wheel with enter a 3

. So R ontains x 1

or x n

and Q,

Q 0

have no adjaent nodes. W.l.o.g. assumethat y 1

isadjaent to x 1

orx n

. Then thereis a

3PC(a 1

a 3

z l

;x 1

) ora 3PC(a 2

a 3

y k

;x n

). 2

Lemma 8.4 Let P =x 1

;:::;x n

bea hordless path inGn suhthat x 1

isof Typet1w.r.t.

, sayadjaent to a 1

, x n

has a neighbor in nfa 1

;a 2

;a 3

g, and no interior node of P has a

neighbor in. Thenone of the following holds.

(i) x n

isof Type t1,p1 or p3 w.r.t. , with a neighbor in P 1

.

(ii) x n

isof Type t2,t2p or t3pw.r.t. and N(x n

)\(P 2

[P 3

)=fb 2

;b 3

g.

(iii) x n

isof Type t2p or t3p w.r.t. with atleast two neighborsin fa 1

;a 2

;a 3

g.

(iv) x n

isof Type p2or p4 w.r.t. and it isadjaent toa 1

.

(v) x n

isof Type t4,t5 or t6 w.r.t. .

Proof: Supposex n

isof Type t1orp1 withauniqueneighboru in. Ifu isnotinP 1

,then

P 2

[P 3

[P[xinduesa3PC(a 1

a 2

a 3

;u). Similarly,ifx n

(22)

xn

x1

_x1

xn

Figure6: Pathsfrom a Type t2 node

(i),elsethere isa 3PC(a 1

a 2

a 3

;x n

). Supposex n

isof Typep2,with neighborsu and v in,

andw.l.o.g. assume thatuand varenotinP 3

. Ifx n

doesnotsatisfy(iv),thenP 1

[P 2

[P

induesa3PC(uvx n

;a 1

). Ifx n

isofTypet2anditdoesnotsatisfy(ii),thenw.l.o.g. wemay

assume that it is adjaent to b 1

and b 3

, and hene P [P 2

[P 3

indues a 3PC(a 1

a 2

a 3

;b 3

).

Supposex n

isof Typet2p ort3pand itdoesnotsatisfy (ii)or(iii). W.l.o.g. x n

is adjaent

tob 1

;b 3

and ithasaneighborinP 2

nb 2

. Then(P[P 1

[P 2

)nb 2

ontainsa3PC(a 1

a 2

a 3

;x n

).

Ifx n

isofTypet3,thenitisadjaentto b 1

;b 2

;b 3

andP[P 1

[P 2

induesa3PC(b 1

b 2

x n

;a 1

).

Finallyassumethat x n

is of Type p4. If theneighborsof x n

in areontainedin P 2

[P 3

,

then((P 2

[P 3

)nfb 2

;b 3

g)[P[a 1

ontainsa3PC(a 1

a 2

a 3

;x n

). Else,wemayassumew.l.o.g.

that the neighbors of x n

in are ontained in P 1

[P 2

. If x n

does not satisfy (v), then

(nfb 2

;a 0 1

g)[P ontainsa 3PC(a 1

a 2

a 3

;x n

). 2

;:::;x n

isof Typet2w.r.t.

, say adjaent to a 1

and a 3

, x n

;a 2

;a 3

g, and no interior node of

P has a neighbor in . Then one of the following holds.

(i) x n

isof Type t1,p1 or p3 w.r.t. , with a neighbor in P 2

.

(ii) x n

isof Type t2,t2p or t3pw.r.t. and N(x n

)\(P 1

[P 3

)=fb 1

;b 3

g.

(iii) x n

;a 2

;a 3

g.

(iv) x n

(23)

n

hold. Thenw.l.o.g. uisinP 1

andheneP 1

[P 3

[P induesa3PC(x 1

a 1

a 3

;u). Supposex n isofTypep3and itdoesnotsatisfy(i). W.l.o.g. theneighborsofx

n

inareinP 1

. Ifn=2

and x n

is adjaent to a 1

,then P 1

[P 2

[P [a 3

ontains an odd wheel withenter a 1

, and

otherwiseP 1

[P 3

[P ontainsa3PC(x 1

a 1

a 3

;x n

). Supposex n

isofTypep2,withneighbors

uandvin,andw.l.o.g. assumethatuandvarenotinP 3

. Ifx n

isnotadjaenttoa 1

,then

P 1

[P 2

[P induesa 3PC(x n

uv;a 1

). So x n

is adjaent to a 1

,and hene P 1

[P 2

[P [a 3 indues an odd wheel with enter a

1

when n > 2 and P 1

[P 3

[P indues an odd wheel

withenter a 1

when n=2. Ifx n

is ofType t2,adjaent to b 2

andsay b 1

,thenP 1

[P 2

[P

indues a3PC(x n

b 1

b 2

;a 1

). So, ifx n

is of Typet2, thenit satises(ii). Similarly,ifx n

is of

Typet3,then thereisa 3PC(x n

b 1

b 2

;a 1

). Ifx n

isof Typet2port3p,and itdoesnotsatisfy

(ii) or(iii),then w.l.o.g. we may assumethat x n

isadjaent to b 1

;b 2

and a node of P 3

nb 3

.

But then P 1

[P 2

[P indues a 3PC(x n

b 1

b 2

;a 1

). Finallyassume that x n

isof Type p4. If

theneighborsof x n

inareontainedinP 1

[P 3

,thenw.l.o.g. x n

isnotadjaentto a 3

and

hene(nfa 1

;a 0 3

g)[P ontainsa3PC(b 1

b 2

b 3

;x n

). Otherwise,w.l.o.g. wemayassumethat

the neighbors of x n

in are ontained inP 1

[P 2

, and hene (nfa 1

;a 2

g)[P ontains a

3PC(b 1

b 2

b 3

;x n

) 2

;:::;x n

isofTypet3w.r.t.

, say adjaent to a 1

;a 2

and a 3

, x n

;a 2

;a 3

g, and no interior node

of P has a neighbor in . Then one of the following holds.

(i) x n

isof Type p2or t3 w.r.t. .

(ii) n=2, x n

is of Type t2p or t3p w.r.t. with at least two neighbors in fb 1

;b 2

;b 3

g and

x n

isadjaent to a 1

;a 2

or a 3

.

(iii) x n

;a 2

;a 3

g.

(iv) n=2, x n

is of Typep3 w.r.t. adjaent to a 1

,a 2

or a 3

.

(v) x n

isof Type t4,t5 or t6 w.r.t .

Proof: Ifx n

isofTypet1otp1withitsneighboruinsayP 1

,thenthere isa3PC(a 1

a 2

x 1

;u).

If x n

is of Type p3, with neighbors in say P 1

, and (iv) does not hold, then P 2

[P 3

[P

ontainsa 3PC(x 1

a 1

a 2

;x n

). By Lemma8.5, x n

annot beof Type t2 w.r.t. . Supposex n is of Typet2p ort3p, but(ii) and (iii)do not hold. Then w.l.o.g. x

n

is a siblingof b 1

, and

hene(P 1

[P 2

[P)nb 1

ontainsa3PC(x 1

a 1

a 2

;x n

). Finallyassume thatx n

is ofTypep4,

withneighborsw.l.o.g. inP 2

[P 3

. Then ([P)nfa 1

;a 3

gontainsa 3PC(b 1

b 2

b 3

;x n

(24)

Theorem 9.1 Let Gbe a square-freeeven-signable graph. If G ontains a =3PC(;)

witha Type t4, t5or t6b node then G has a star utset.

Proof: AssumeGhasnostarutset. ThenbyTheorem4.6, Gontainsnoproperwheelthat

isnot abeetle.

Let C be theset ofall ordered pairs;u suh that=3PC(;)and u is of Type t4,

t5 ort6bw.r.t. .

For;u 2 C, we assume w.l.o.g. that ifu is of Type t5 w.r.t. then u is not adjaent

to a 3

,ifu isofTypet4dw.r.t. then uisnotadjaentto a 3

and b 2

,andifuis ofTypet4s

w.r.t. thenu isnotadjaent to a 3

and b 3

.

For ;u 2 C dene the orresponding sets S as follows. If u is of Type t5 or t6 w.r.t.

, then let S = (N(u) [u) n( n fa 1

;a 2

;b 2

;b 3

g). If u is of Type t4d w.r.t. , then

let S = (N(u) [u) n( n fa 1

;a 2

;b 1

;b 3

g). If u is of Type t4s w.r.t. , then let S =

(N(u)[u)n(nfa 1

;a 2

;b 1

;b 2

g). SineS isnota starutset,there existsadiretonnetion

P =x 1

;:::;x n

inGnSfromP 1

[P 2

toP 3

. Let;ubehosenfromC sothattheardinality

of N(u)\ isminimizedand,subjetto this, thesizeoftheorrespondingP is minimized.

Claim 1: Nonode of P isof Type t4, t5or t6 w.r.t. .

Proof of Claim 1: It isenoughto showthatifv andwarebothofTypet4,t5ort6w.r.t. ,

thenvwis anedge. Supposenot. Sine v and wbothhaveat least twoneighborsineah of

thesets fa 1

;a 2

;a 3

gand fb 1

;b 2

;b 3

g,forsomei;j2f1;2;3g, a i

and b j

are ommonneighbors

of v and w. Sine fa i

;b j

;u;vg annot induea square, a i

b j

is an edge. W.l.o.g. i=j =2,

N(v)\fa 1

;a 2

;a 3

g=fa 1

;a 2

gandN(w)\fa 1

;a 2

;a 3

g=fa 2

;a 3

g. Butthenfa 1

;a 2

;a 3

;b 2

;v;wg

induesan oddwheel withenter a 2

. This ompletestheproofof Claim1.

Claim 2: Nonode of P isof Type p3w.r.t. .

Proof of Claim 2: Suppose x i

is of Type p3 w.r.t. . Let 0

be obtained from by

substitutingx i

into. Sinex i

2GnS,nodex i

isnotadjaenttou. ThereforejN(u)\ 0

j

jN(u)\j. Therefore 0

;uandP 0

,whereP 0

=P x

1 x

i 1 orP

0 =P

x i+1

x n

,ontraditourhoie

of ;u andP. Thisompletes theproof ofClaim 2.

Claim 3: Nonode of P isof Type t2p or t3p w.r.t. .

Proof of Claim 3: Suppose that x i

is of Type t2p or t3p w.r.t. and let 0

be obtained

from by substitutingx i

for its sibling. Note that u annot be of Type t6 w.r.t. , sine

otherwiseu is of Type t5 w.r.t. 0

, ontraditing ourhoie of ;u. In partiular, u is not

adjaent toa 3

.

Supposex i

is a siblingof a 1

. Sine u is adjaent to a 2

;b 1

and at leastone of b 2

;b 3

, and

it is notadjaent to x i

and a 3

,node u must be of Type t2por t3pw.r.t 0

,adjaent to b 3

.

In partiular, N(u)\P 3

=fb 3

g and b 1

is the unique neighbor of u in the x i

b 1

-path of 0

.

Node x i

is not adjaent to b 1

, else fa 2

;b 1

;x i

;ug indues a square. Hene (H 13

;u) must be

a line wheel with u adjaent to a 0 1

. But then (P 1

nfa 1

;b 1

g)[P 3

[fa 2

;u;x i

g ontains a

3PC(x i

a 2

a 3

(25)

i 2 1 1 2 3 isnotadjaentto a

3 andx

i

,itmustbeof Typet3pw.r.t. 0

,andheneofTypet5w.r.t. .

Sine uis ofType t3pw.r.t. 0

,N(u)\P 3

=fb 3

g. But thisontradits Lemma6.2applied

to and u.

Supposex i

isasiblingof a 3

. Theni>1andu isofthesametype w.r.t. 0

asitisw.r.t.

. Butthen 0

;u andP x1x

i 1

ontradit ourhoie of ;uand P.

Supposex i

isa siblingofb 1

. Thenu isof Type t2port4dw.r.t. 0

,adjaent to b 3

. If u

is of Typet4d w.r.t. 0

,then 0

;u ontradit ourhoie of ;u. So u is of Type t2p w.r.t.

0

. In partiular, N(u)\P 2

=fa 2

g. So u must be of Type t4d w.r.t. . Node x i

annot

be adjaent to a 1

, elsefa 1

;b 3

;x i

;ug indues a square. So a 1

b 1

is not an edge. By Lemma

6.2, N(u)\P 3

= fb 3

g and u has a neighbor in P 1

nfa 1

;b 1

g. So there is a subpath P 0 of P 1 nfa 1 ;b 1

g suh that one endnode of P 0

is adjaent to u, the other to x i

and no proper

subpathof P 0

hasthisproperty. Butthen P 2 [P 0 [fb 3 ;x i

;ug induesa 3PC(b 2 b 3 x i ;u). Suppose x i

is a sibling of b 2

. Then u must be of Type t4d w.r.t. 0

. Node x i

is not

adjaent to a 2

,else fa 2

;b 3

;x i

;ug induesa square. So i=1. Sine u isadjaent to b 3

,node

b 3

isinS andhene n>1. Butthen 0

;u and P x

2 x

n

ontradit ourhoie of;u and P.

Finallysupposethatx i

isa siblingofb 3

. Then uis ofTypet4sw.r.t. 0

. Ifu isofType

t5w.r.t. ,then 0

;uontraditourhoieof;u. SouisofTypet4sw.r.t. . Henei=n

and n>1. But then 0

;u and P x

1 x

i 1

ontradit ourhoie of ;u and P. This ompletes

theproofof Claim3.

Claim 4: If x i

isof Typep4w.r.t. , then i=1 and the neighbors of x i

in are ontained

in P 1

[P 2

.

Proof of Claim4: Supposex i

isofTypep4w.r.t. . Iftheneighborsofx i

inareontained

inP 1

[P 2

then i=1.

Supposethattheneighborsofx i

inareontainedinP k

[P 3

fork=1or2. Forj =k;3,

letu j

(resp. v j

) betheneighborofx i

isP j

thatislosesttoa j

(resp. b j

). IfuisofTypet6b

w.r.t. , thenbyLemma6.2,uisadjaenttoall nodesofandhenefu;x i ;v k ;u 3 gindues

a square. Therefore, u isnot adjaent to a 3

.

Firstsupposethatx i

isadjaenttoa 3

. Thenx i

isnotadjaenttoa k

andso([x i )nP 3 v3b3 indues a 0

= 3PC(a 1 a 2 a 3 ;u k v k x i

). Sine u is adjaent to a 1

;a 2

;b 1

and it is not adjaent

to a 3

;x i

,it mustbeof Typet4sw.r.t. 0

. Heneu isadjaent to u k

and v k

. Node u annot

have neighbors in P 3

, sine otherwise 0

;u would ontradit the hoie of ;u. So u is of

Typet4s w.r.t. . Sine uis adjaent to u k

6=a k

, (H 12

;u)must bea universalwheel. Then

byLemma 6.2, a 1 b 1 and a 2 b 2

are notedges and soboth(H 13

;u)and (H 23

;u)mustbe twin

wheels. Hene both P 1

and P 2

are of length 2. But then fa k ;a 3 ;u k ;x i

g indues a square.

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 k v

k b

k

. Sineuisadjaenttoa 1

;a 2

and

atleastoneofb 2

;b 3

(i.e. ithasaneighborinthea 3

v 3 k

-pathof 0

),anditisnotadjaent to

a 3

andx i

,itmustbeofTypet4dw.r.t. 0

. Ifk =1, 0

;uontraditsourhoieof;usine

b 1

is adjaent to u and belongs to n 0

. So k = 2. If a 1

b 1

is an edge, then fu;x i ;u 3 ;b 1 g

induesasquare. ButthenbyLemma6.2,v 3

=b 3

andN(u)\=fa 1 ;a 2 ;b 1 ;b 3 ;b 0 3 g. Hene P 2

[fu;x i ;b 1 ;b 0 3

g induesa 3PC(u 2

v 2

x i

;u). Thisompletes theproofof Claim4.

Claim 5: n> 1, x 1

is of Type t1, p1, p2, t2, t3 or p4 w.r.t. , and x n