Clique width and Well Quasi Ordering of Triangle Free graph classes

Contents lists available atScienceDirect

Journal

of

Computer

and

System

Sciences

www.elsevier.com/locate/jcss

Clique-width

and

well-quasi-ordering

of

triangle-free

graph

classes

✩

Konrad K. Dabrowski

a

,

∗

,

Vadim V. Lozin

b

,

Daniël Paulusma

a

a_{DepartmentofComputerScience,DurhamUniversity,LowerMountjoy,SouthRoad,DurhamDH13LE,UnitedKingdom} b_{MathematicsInstitute,UniversityofWarwick,CoventryCV47AL,UnitedKingdom}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received24May2018

Receivedinrevisedform 29December2018 Accepted9September2019

Availableonlinexxxx

Keywords:

Clique-width

Forbiddeninducedsubgraph Hereditarygraphclass Well-quasi-ordering

We obtainacompleteclassificationofgraphs Hforwhichtheclassof

(

triangle, H)-free graphs is well-quasi-orderedby the inducedsubgraph relationand analmostcomplete classification of graphs H for which the classof (triangle, H)-free graphs has bounded clique-width. Inparticular, weshow thatforthese graphclasses,well-quasi-orderability implies boundedness ofclique-width. To obtain our results,we furtherrefine aknown methodbasedoncanonicaldecomposition.Thisleadstoanewdecompositiontechnique thatisapplicabletobothnotions,well-quasi-orderabilityandclique-width.

©2019TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCC BYlicense(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Clique-width.EveryNP-hardgraphproblembecomespolynomial-timesolvableafterplacingappropriaterestrictionsonthe input.Ageneralmethodinthedesignofgraphalgorithms forspecialgraphclassesistodecompose thevertexsetofthe graphintolargesets of“similarlybehaving”’verticesandtoexploitthisdecompositionalgorithmically.An optimalvertex set decompositiongivesusthe “width” ofthe graph.Clique-width isone ofthe moststudied widthparameters (see, for example,thesurveys [26,29]).Thevertexsetdecompositionofclique-widthis definedvia agraphconstruction basedon vertexlabellings.Startingfromtheemptygraph,agraphG isbuiltupvertex-by-vertexusingfourspecificgraphoperations. Theseoperationsensurethatverticeslabelledalikewillkeepthesamelabelandthus“behave”identically.Theclique-width ofG istheminimumnumberofdifferentlabelsneededtoconstructG inthisway(seeSection 2foraprecisedefinition). A graphclass

G

hasboundedclique-widthifthereexistsaconstantcsuchthateverygraphin

G

hasclique-widthatmostc. Thealgorithmicimportanceofhavingboundedclique-widthfollowsfromtheexistenceofseveralmeta-theorems [10,21,31, 41] which,whencombinedwithanapproximationresult [39],ensurethatmanywell-knownNP-hardgraphproblems,such asGraphColouringandHamiltonCycle,becomepolynomial-timesolvableforeverygraphclassofboundedclique-width. Hence,thereisaneedtoverifyboundednessofclique-widthofspecialgraphclasses,inparticularwhenundertakinga sys-tematicclassificationintothecomputationalcomplexity ofgraphproblemsunderinput restrictions;see,forexample, [25] fortheimportanceofclique-widthfortheGraphColouringproblem.

✩ _{ThisresearchwassupportedbyEPSRC(EP/K025090/1andEP/L020408/1)andtheLeverhulmeTrustRPG-2016-258.Anextendedabstractofthispaper}

appearedintheproceedingsofWG2017 [14].

*

Correspondingauthor.

E-mailaddresses:[email protected](K.K. Dabrowski),[email protected](V.V. Lozin),[email protected](D. Paulusma).

https://doi.org/10.1016/j.jcss.2019.09.001

Well-quasi-orderings.Agraphclass

G

iswell-quasi-orderedbyacontainmentrelationifforanyinfinitesequenceG0

,

G1

,

. . .

of graphs in

G

, there is a pair i,j with i

<

j such that Gi is contained in Gj. Justas is the case for having bounded clique-width, beingwell-quasi-orderedisahighlydesirableproperty,whichhasbeenfrequentlydiscoveredintheareasof discrete mathematicsand theoreticalcomputer science [1,23,35].To illustrateits importance, let usmentiontheseminal project ofRobertsonandSeymour ongraph minors,whichculminated in2004in theproof [43] ofWagner’s conjecture. Wagner’sconjecturestatesthatthesetofallfinitegraphsiswell-quasi-orderedbytheminorrelation(agraphHisaminor ofagraphGif HcanbeobtainedfromG viaasequenceofvertexdeletions,edgedeletionsandedgecontractions).Asan algorithmicconsequence,givenaminor-closedgraphclass,itispossibletotestincubictimewhetheragivengraphbelongs tothisclass [42] (see [28] foraquadraticalgorithm).

Togivesomemoreexamples,aresultofDing [20] impliesthateveryclassofgraphswithboundedvertexcovernumber iswell-quasi-orderedbytheinducedsubgraphrelation,whereasMader [38] showedthateveryclassofgraphswithbounded feedbackvertexnumberiswell-quasi-orderedbythetopologicalminorrelation.Fellows,HermelinandRosamund [22] sim-plifiedtheproofsoftheseresultsandalsoshowedthateveryclassofgraphsofboundedcircumferenceiswell-quasi-ordered by the induced minorrelation.Furthermore,as “interestingapplications”of thesethreeresults, they gavelinear-time al-gorithms forrecognizing graphs from anytopological-minor-closed graph class of boundedfeedback vertexnumber, any induced-minor-closedgraphclassofboundedcircumference,andanyinduced-subgraph-closedgraphclassofbounded ver-texcovernumber.

Hereditarygraphclasses. Graph classes closed under taking induced subgraphs are said to be hereditary. Courcelle [9] proved thattheclassofgraphsobtainedfromgraphsofclique-width 3 viaone ormoreedgecontractionshasunbounded clique-width.Thismeansthattheclique-widthofagraphcanbemuchsmallerthantheclique-widthofitsminors.Onthe otherhand,theclique-widthofagraphisatleasttheclique-widthofanyofitsinducedsubgraphs(see,forexample, [11]). Hence,itisnaturaltofocusondeterminingboundednessofclique-widthforhereditarygraphclasses.

Researchgoals. Nothing in the definitions of clique-widthand well-quasi-orderability suggeststhat there is anything in common betweenthetwo notions. However, aswe willdiscuss, recent resultsforhereditarygraph classessuggestedan intriguing connection betweenthem. This connection is not yet well understood,and as such, it remains to be further explored.Ourunderlyingresearchgoalsare:

– toincreaseunderstandingoftherelationbetweenwell-quasi-orderabilitybytheinducedsubgraphrelationand bound-ednessofclique-widthforhereditarygraphclasses;and

– toobtainnewresultsforbothnotionsappliedtohereditarygraphclasses.

Inaprevious paper [15] weshowedthatcertain graphoperationsandgraphconstructionsworkequallywellforbounded clique-widthandwell-quasi-orderability.Inthispaperwewillexplorecommongraphtechniquesfurther.Beforediscussing ourresultsindetail,wefirstdiscussaconjecturethatmotivatedourresearch.

Conjecture.Wefirstnotethatthehereditaryclassofgraphsofdegreeatmost 2 isnotwell-quasi-orderedbytheinduced subgraph relation,asit containsthe classofcycles,whichformaninfiniteantichain. Asevery graphofdegreeatmost 2 hasclique-widthatmost 4,havingboundedclique-widthdoesnotimplywell-quasi-orderabilitybytheinducedsubgraph relation.In2010,Daligault,RaoandThomassé [18] askedaboutthereverseimplication:doeseveryhereditarygraphclass that iswell-quasi-orderedby theinduced subgraphrelationhaveboundedclique-width? AtWG 2015, Lozin, Razgonand Zamaraev [37] gavea negative answer tothisquestion. It isreadily seen that aclass ofgraphs ishereditaryif andonly ifitcan becharacterisedbya uniqueset

F

ofminimalforbiddeninduced subgraphs.As theset

F

ofminimal forbidden inducedsubgraphsinthecounter-exampleof [37] isinfinite,thequestionofDaligault,RaoandThomassé [18] remainsopen forfinitelydefinedhereditarygraphclasses,thatis,hereditarygraphclassesforwhich

F

isfinite.

Conjecture1([37]).Ifafinitelydefinedhereditaryclassofgraphs

G

iswell-quasi-orderedbytheinducedsubgraphrelation,then

G

hasboundedclique-width.

IfConjecture1weretrue,thenforfinitelydefinedhereditarygraphclasses,theaforementionedalgorithmicconsequences of having bounded clique-widthwould also holdfor the property of beingwell-quasi-ordered by the induced subgraph relation.Infact,beingwell-quasi-orderedcouldbetheunderlyingpropertyenablingtheseconsequences.

A hereditarygraphclass definedbya single forbiddeninducedsubgraph H iswell-quasi-orderedby theinduced sub-graphrelationifandonlyifithasboundedclique-widthifandonlyifHisaninducedsubgraphofP4(see,forinstance, [17,

19,33]).1 HenceConjecture1holdswhen

F

hassize 1.Weconsiderthecasewhen

F

hassize 2,say

F

= {

H1

,

H2

}

.Such

graph classesaresaidto bebigenicor

(H

1

,

H2

)-free

graphclasses.InthiscaseConjecture1isalsoknownto betrue

ex-ceptfortwostubbornopencases,namely

(H

1

,

H2

)

=

(K

3

,

P2

+

P4

)

and

(H

1

,

H2

)

=

(P

1

+

P4

,

P2

+

P3

);

see [15].Inbothof

[image:3.561.120.424.52.127.2]

Fig. 1.The forbidden induced subgraphs considered in our results.

thesecases,itisneitherknownwhetherthegraphclassunderconsiderationhasboundedclique-widthnorwhetheritis well-quasi-orderedbytheinducedsubgraphrelation.

Resultsandmethodology. In Section 5, we prove that Conjecture1 holds for the class of

(K

3

,

P2

+

P4

)-free

graphs by

showingthat thisclasshas boundedclique-widthandiswell-quasi-orderedby theinduced subgraph relation.Insteadof using ad hocarguments, we emphasize that our underlyingresearch goalis to develop methodology that could also be applied morewidely toother classesandwhich increasesourunderstanding ofthe structureof specialgraphclasses.In thispaperwefocusonK3-freegraphsandweprovethetworesultson

(K

3

,

P2

+

P4

)-free

graphsbyusingageneralmethod

explainedinSection4.

Ourmethodisbasedonextending(alabelledversionof)well-quasi-orderabilityorboundednessofclique-widthofthe bipartite graphsinahereditarygraphclass X toa specialsubclassof3-partitegraphsin X.Thecrucialpropertyofthese 3-partitegraphsisthatnothreeverticesfromthethreedifferentpartitionclassesformacliqueorindependentset.Wesay thatsuch 3-partitegraphsarecurious.ThemethodfindsitsorigininapaperofFouquet,GiakoumakisandVanherpe [24], who characterized totally decomposablebipartite graphs, that is,bipartite graphs that can recursively be canonically de-composedintographsisomorphictoanisolatedvertex.Dabrowski,DrossandPaulusma [12] generalizedthisdecomposition frombipartite graphstok-partitegraphsforanyk

≥

2 andpartiallycharacterizedtotally3-decomposablegraphs.This suf-ficedtoprovethat

(K

3

,

P1

+

P5

)-free

graphshaveboundedclique-width [12],butitisnotstrongenoughforourpurposes.

Ournewdecompositionmethodallowsustoalsodealwithcuriousgraphsthatarenottotally3-decomposable.

In Section 5 we show how to generalize results for curious

(K

3

,

P2

+

P4

)-free

graphs to the whole class of

(K

3

,

P2

+

P4

)-free

graphs.Duetothegeneralityofourmethod,itimmediatelyfollowsthattheclassof

(K

3

,

P1

+

P5

)-free

graphs is also well-quasi-ordered, which was not previously known (see [3,33]). Since

(K

3

,

P1

+

2P2

)-free

graphs are

(K

3

,

P1

+

P5

)-free

(and

(K

3

,

P2

+

P4

)-free),

our results immediately imply that the class of

(K

3

,

P1

+

2P2

)-free

graphs

is also well-quasi-ordered by the induced subgraph relation, which was also previously unknown (see [3,33]).As men-tioned,it wasalreadyknown [12] thattheclassof

(K

3

,

P1

+

P5

)-free

graphshasboundedclique-width(andsothesame

wasknownfortheclassof

(K

3

,

P1

+

2P2

)-free

graphs);aswewillsee,thisalsofollowsdirectlyfromourmethod.SeeFig.1

forpicturesoftheforbiddeninducedsubgraphsmentionedinthisparagraph.

DichotomyforK3-freegraphs. Previously, well-quasi-orderability was known for

(K

3

,

P6

)-free

graphs [3],

(P

2

+

P4

)-free

bipartitegraphs [32] and

(P

1

+

P5

)-free

bipartitegraphs [32]. Ithasalsobeenshownthat H-freebipartite graphsarenot

well-quasi-orderedif H contains an induced3P1

+

P2 [33], 3P2 [20] or 2P3 [32].Moreover, forevery s

≥

1,theclass of

(K

3

,

s P1

)-free

graphsisfiniteduetoRamsey’sTheorem [40].Theaboveresultsleadtothefollowingknowndichotomyfor

H-freebipartitegraphs.

Theorem1.Let H beagraph.TheclassofH -freebipartitegraphsiswell-quasi-orderedbytheinducedsubgraphrelationifandonlyif H

=

s P1forsomes

≥

1or H isaninducedsubgraphofP1

+

P5,P2

+

P4or P6.

Now combiningthe aforementioned known resultsfor

(K

3

,

H)-free graphs and H-free bipartite graphs withour new

resultsyieldsthefollowingnewdichotomyfor H-freetriangle-freegraphs,whichisexactlythesameastheone in Theo-rem1.

Theorem2.Let H beagraph.Theclassof

(K

3

,

H)-freegraphsiswell-quasi-orderedbytheinducedsubgraphrelationifandonlyif

H

=

s P1forsomes

≥

1orH isaninducedsubgraphofP1

+

P5,P2

+

P4,or P6.

Besidesourmethodbasedoncuriousgraphs,wealsoexpectthatTheorem2willitselfbeausefulingredientforshowing resultson well-quasi-orderabilityforother graphclasses,justasTheorem 1hasalreadyproven to beusefulforthis(see e.g. [32]).

Forclique-widththefollowingdichotomy isknownfor H-free bipartitegraphs; seeSection 2forthedefinitionofthe graphSh,i,j.

Theorem3([16]).Let H beagraph.TheclassofH -freebipartitegraphshasboundedclique-widthifandonly ifH

=

s P1forsome

Itwouldbeinterestingtodeterminewhether

(K

3

,

H)-freegraphsallowthesamedichotomywithrespecttothe

bound-ednessoftheir clique-width.The evidencesofarisaffirmative,butinordertoanswerthisquestion tworemaining cases need tobesolved, namely

(H

1

,

H2

)

=

(K

3

,

P1

+

S1,1,3

)

and

(H

1

,

H2

)

=

(K

3

,

S1,2,3

).

Both casesturnouttobe highly

non-trivial;inparticular,theclassof

(K

3

,

P1

+

S1,1,3

)-free

graphscontainstheclassof

(K

3

,

P1

+

P5

)-free

graphs,andtheclass

of

(K

3

,

S1,2,3

)-free

graphscontainsboththeclassesof

(K

3

,

P1

+

P5

)-free

and

(K

3

,

P2

+

P4

)-free

graphs.

In Section 6 we give state-of-the-art summaries for well-quasi-orderability and boundedness of clique-width of bi-genic graphclasses (which include ournew results), together with an overviewof the missingcases for both problems (which include the missing cases mentioned in this section). In particular, these summaries show that our results for

(K

3

,

P2

+

P4

)-free

graphs are tightin thesense that

(H,

P2

+

P4

)-free

graphs haveunbounded clique-widthandarenot

well-quasi-orderedbytheinducedsubgraphrelationforanyinducedsupergraphH of K3,unlessH isthepaw(thegraph

P1

+

P3).InSection7weconcludeourpaperwithsomeremarksforfuturework.

2. Preliminaries

Throughoutthepaper,weconsideronlyfinite,undirectedgraphswithoutmultipleedgesorself-loops.Below,wedefine furthergraphterminology.

Thedisjointunion

(V

(G)

∪

V

(H),

E

(G)

∪

E(H))oftwovertex-disjointgraphsGandHisdenotedbyG

+

Handthedisjoint unionofrcopiesofagraphG isdenotedbyrG.Thecomplement G ofagraphG hasvertexset V

(G)

=

V

(G)

andanedge betweentwodistinct verticesu,v ifandonlyifuv

∈

/

E(G).Forasubset S

⊆

V

(G),

welet G

[

S

]

denotethesubgraphofG inducedby S,whichhasvertexsetSandedgeset

{

uv

|

u,v

∈

S,uv

∈

E

(G)

}

.IfS

= {

s1

,

. . . ,

sr}then,tosimplifynotation,we

mayalsowriteG

[

s1

,

. . . ,

sr]insteadofG

[{

s1

,

. . . ,

sr}].WeuseG

\

S todenotethegraphobtainedfromGbydeletingevery

vertexinS,thatis,G

\

S

=

G

[

V

(G)

\

S

]

.Wewrite G

⊆i

Gtoindicatethat G isaninducedsubgraphofG.

The graphs Cr, Kr, K1,r−1 and Pr denote the cycle, complete graph, star and path on r vertices, respectively. The graphs K3 and K1,3 are also called the triangle and claw, respectively. A graph G is a linearforest ifevery component

ofG isapath(on atleastonevertex).ThegraphSh,i,j,for1

≤

h

≤

i

≤

j,denotesthesubdividedclaw,thatis,thetreethat hasonlyone vertexxofdegree 3 andexactlythree leaves,whichare atdistanceh, i and j fromx,respectively.Observe that S1,1,1

=

K1,3.Welet

S

denotetheclassofgraphs,eachconnectedcomponentofwhichiseitherasubdividedclawor

apath.

Forasetofgraphs

{

H1

,

. . . ,

Hp},agraphGis

(H

1

,

. . . ,

Hp)-freeifithasnoinducedsubgraphisomorphictoagraphin

{

H1

,

. . . ,

Hp};ifp

=

1,wemaywrite H1-freeinsteadof

(H

1

)-free.

Agraphisk-partiteifitsvertexcanbepartitionedintok

(possibly empty)independent sets; 2-partite graphsare also knownas bipartite graphs.The biclique orcompletebipartite graph Kr,s isthebipartitegraphwithsetsinthepartitionofsizerandsrespectively,suchthat everyvertexinonesetis adjacenttoeveryvertexintheotherset.ForagraphG

=

(V

,

E),thesetN(u)

= {

v

∈

V

|

uv

∈

E

}

denotestheneighbourhood of u

∈

V.Let X be aset ofvertices in G.Avertex y

∈

V

\

X iscomplete to X ifit is adjacentto every vertexof X and anti-complete to X ifitis non-adjacentto everyvertexof X. Asetofvertices Y

⊆

V

\

X iscomplete (resp.anti-complete) to X ifevery vertexin Y is complete(resp. anti-complete)to X.If X and Y are disjointsets ofvertices ina graph,we saythat theedges betweenthesetwosets forma matching ifeach vertexin X hasatmostone neighbourinY andvice versa;ifeachvertexhasexactlyonesuchneighbour,wesaythatthematchingisperfect.Similarly,theedgesbetweenthese sets form aco-matching ifeach vertexin X has atmostone non-neighbourin Y andviceversa. We saythat theset X dominates Y ifeveryvertexofY hasatleastone neighbourin X.Similarly,avertexx dominates Y ifevery vertexofY is adjacenttox.Avertexy

∈

V

\

X distinguishes Xify hasbothaneighbourandanon-neighbourinX.Theset X isamodule ofGifnovertexinV

\

X distinguishes X.Amodule X isnon-trivialif1

<

|

X

|

<

|

V

|

,otherwiseitistrivial.Agraphisprime ifithasonlytrivialmodules.Twoverticesarefalsetwinsiftheyhavethesameneighbourhood(notethatsuchverticesmust benon-adjacent).Clearlyanyprimegraphonatleastthreeverticescannotcontainapairoffalsetwins,asanysuchpairof verticeswouldformanon-trivialmodule.

Wewillusethefollowingstructuralresult.

Lemma1([12]).Let G beaconnected

(K

3

,

C5

,

S1,2,3

)-free

graphthatdoesnotcontainapairoffalsetwins.Then G iseitherbipartite

oracycle.

2.1. Clique-width

The clique-widthcw(G) ofagraph G isthe minimumnumberoflabelsneededto constructG byusingthe following fouroperations:

1. i(v):creatinganewgraphconsistingofasinglevertexv withlabeli; 2. G1

⊕

G2:takingthedisjointunionoftwolabelledgraphsG1andG2;

We say that aconstruction of a graph G withthe fouroperationsis a k-expressionifit uses atmostk labels.Thus the clique-widthofGistheminimumkforwhichGhasak-expression.Recallthataclassofgraphs

G

hasboundedclique-width ifthereisaconstantcsuchthattheclique-widthofeverygraphin

G

isatmostc;otherwisetheclique-widthisunbounded. LetG be agraph.Wedefine thefollowingoperations. Foraninduced subgraph G

⊆i

G,thesubgraphcomplementation operation(acting onG withrespectto G) replacesevery edgepresentinG byanon-edge,andviceversa.Similarly, for two disjointvertexsubsets S and T in G,the bipartitecomplementation operationwithrespect to S and T acts on G by replacingeveryedgewithoneend-vertexin SandtheotheroneinT byanon-edgeandviceversa.

Wenowstatesomeusefulfactsabouthowtheaboveoperations(andsomeotherones)influencetheclique-widthofa graph.Wewillusethesefactsthroughoutthepaper.Letk

≥

0 beaconstant andlet

γ

besome graphoperation.Wesay thatagraphclass

G

is

(k,

γ

)-obtained

fromagraphclass

G

ifthefollowingtwoconditionshold:

1. everygraphin

G

isobtainedfromagraphin

G

byperforming

γ

atmostktimes,and

2. foreveryG

∈

G

thereexistsatleastonegraphin

G

obtainedfromG byperforming

γ

atmostktimes.

Wesaythat

γ

preservesboundednessofclique-widthifforanyfiniteconstantkandanygraphclass

G

,anygraphclass

G

thatis

(k,

γ

)-obtained

from

G

hasboundedclique-widthifandonlyif

G

hasboundedclique-width.

Fact 1.Vertexdeletionpreservesboundednessofclique-width [36].

Fact 2. Subgraphcomplementationpreservesboundednessofclique-width [29]. Fact 3. Bipartitecomplementationpreservesboundednessofclique-width [29].

Lemma2([11]).Let G beagraphandlet

P

bethesetofallinducedsubgraphsof G thatareprime.Thencw(G)

=

maxH_∈Pcw(H).

2.2. Well-quasi-orderability

Aquasiorder

≤

onaset X isa reflexive,transitivebinaryrelation.Twoelements x,y

∈

X inthisquasi-orderare com-parable if x

≤

y or y

≤

x,otherwise they are incomparable. Aset ofelements in a quasi-orderis achain ifevery pair of elementsiscomparableanditisanantichainifeverypairofelementsisincomparable.Thequasi-order

≤

isa well-quasi-order if anyinfinitesequence of elements x1

,

x2

,

x3

,

. . .

in X contains a pair

(x

i,xj) with xi

≤

xj andi

<

j.Equivalently, a quasi-orderisawell-quasi-order ifandonlyifithasnoinfinite strictlydecreasingsequence x1

x2

x3

· · ·

andno

infiniteantichain.

Foran arbitraryset M, let M∗ denote theset offinite sequences ofelements of M. Aquasi-order

≤

on M definesa quasi-order

≤

∗ on M∗ asfollows:

(a

1

,

. . . ,

am)

≤

∗

(b

1

,

. . . ,

bn)ifandonlyifthereisasequence ofintegers i1

,

. . . ,

im with

1

≤

i1

<

· · ·

<

im

≤

nsuchthataj

≤

bij for j

∈ {

1,

. . . ,

m

}

.Wecall

≤

∗thesubsequencerelation.

Lemma3(Higman’sLemma [27]).If

(M,

≤

)

isawell-quasi-orderthen

(M

∗

,

≤

∗

)

isawell-quasi-order.

Todefine thenotionoflabelledinducedsubgraphs,letusconsideranarbitraryquasi-order

(W

,

≤

).

Wesaythat Gisa labelledgraphifeach vertexv ofG isequippedwithan elementlG(v)

∈

W (the labelofv).Giventwolabelledgraphs G and H,we saythat G isalabelledinducedsubgraphof H ifG is isomorphictoan inducedsubgraphof H andthereisan isomorphismthatmapseachvertexv ofGtoavertexwofH withlG(v)

≤

lH

(w).

Clearly,if

(W

,

≤

)

isawell-quasi-order, thena graphclass X cannot containaninfinitesequenceoflabelledgraphsthat isstrictly-decreasingwithrespecttothe labelled induced subgraph relation.We thereforesay that a graph class X is well-quasi-orderedby the labelled induced subgraphrelationifitcontainsnoinfiniteantichainsoflabelledgraphswhenever

(W

,

≤

)

isawell-quasi-order.Suchaclass isreadilyseentoalsobewell-quasi-orderedbytheinducedsubgraphrelation.

Daligault,RaoandThomassé [18] showedthateveryhereditaryclassofgraphsthatiswell-quasi-orderedbythelabelled induced subgraph relationis definedby a finiteset offorbidden induced subgraphs. Korpelainen, LozinandRazgon [34] conjecturedthatifahereditaryclassofgraphs

G

isdefinedbyafinitesetofforbiddeninducedsubgraphs,then

G

is well-quasi-ordered by the induced subgraph relation ifand only ifit is well-quasi-orderedby the labelled induced subgraph relation.Brignall,EngenandVatter [8] recentlyfoundaclass

G

∗ with14forbiddeninducedsubgraphsthatisa counterex-ampleforthisconjecture,thatis

G

∗ iswell-quasi-orderedbytheinducedsubgraphrelationbutnotbythelabelledinduced subgraphrelation.However,sofar,allknownresultsforbigenicgraphclasses,includingthoseinthispaper,agreewiththe conjectureforbigenicgraphclasses.

Lemma4([15]).Thefollowingoperationspreservewell-quasi-orderabilitybythelabelledinducedsubgraphrelation:

(i) Subgraphcomplementation, (ii) Bipartitecomplementationand (iii) Vertexdeletion.

Lemma5([3]).Ahereditaryclass X ofgraphsiswell-quasi-orderedbythelabelledinducedsubgraphrelationifandonlyifthesetof primegraphsin X is.Inparticular,X iswell-quasi-orderedbythelabelledinducedsubgraphrelationifandonlyifthesetofconnected graphsin X is.

Lemma6([3,32]).

(P

7

,

S1,2,3

)-free

bipartitegraphsarewell-quasi-orderedbythelabelledinducedsubgraphrelation.

Let

(L

1

,

≤

1

)

and

(L

2

,

≤

2

)

bewell-quasi-orders.We definetheCartesianProduct

(L

1

,

≤

1

)

×

(L

2

,

≤

2

)

ofthese

well-quasi-orders asthe order

(L,

≤L

)

on the set L

:=

L1

×

L2 where

(l

1

,

l2

)

≤L

(l

1

,

l2

)

if andonlyifl1

≤

1l1 andl2

≤

2l2. Lemma3

impliesthat

(L,

≤L

)

isalsoawell-quasi-order. IfG hasa labellingwithelements ofL1 andofL2,sayl1

:

V

(G)

→

L1 and

l2

:

V

(G)

→

L2,wecanconstructthecombinedlabellingin

(L

1

,

≤

1

)

×

(L

2

,

≤

2

)

that labelseachvertexv ofG withthelabel

(l

1

(v),

l2

(v)).

Lemma7.Fixawell-quasi-order

(L

1

,

≤

1

)

thathasatleastoneelement.Let X beaclassofgraphs.ForeachG

∈

X fixalabellingl1_G

:

V

(G)

→

L1.Then X iswell-quasi-orderedbythelabelledinducedsubgraphrelationifandonlyifforeverywell-quasi-order

(L

2

,

≤

2

)

andeverylabellingofthegraphsin X bythisorder,thecombinedlabellingin

(L

1

,

≤

1

)

×

(L

2

,

≤

2

)

obtainedfromtheselabellingsalso

resultsinawell-quasi-orderedsetoflabelledgraphs.

Proof. IfX iswell-quasi-orderedbythelabelledinducedsubgraphrelationthenbydefinitionitiswell-quasi-orderedwhen labelledwithlabelsfrom

(L

1

,

≤

1

)

×

(L

2

,

≤

2

).

If X isnotwell-quasi-orderedbythelabelledinducedsubgraphrelationthen

theremustbeawell-quasi-order

(L

2

,

≤

2

)

andaninfinitesetofgraphsG1

,

G2

,

. . .

whoseverticesarelabelledwithelements

of L2 such that thesegraphs form an infinite labelledantichain. Foreach graph Gi, replace the label l on vertex v by

(l

1_G1

(v),

l).The graphsarenowlabelledwithelementsofthewell-quasi-order

(L

1

,

≤

1

)

×

(L

2

,

≤

2

)

andresultinaninfinite

labelledantichainofgraphslabelledwithsuchcombinedlabellings.Thiscompletestheproof.

2

2.3. k-uniformgraphs

For an integerk

≥

1, a graph G is k-uniform ifthere isa symmetric square 0,1 matrix K oforder k anda graph Fk on vertices1,2,

. . . ,

k such thatG

∈

P

(K

,

Fk),where

P

(K

,

Fk)isthegraphclassdefinedasfollows.Let H bethedisjoint union ofinfinitelymanycopiesof Fk.Fori

∈ {

1,

. . . ,

k

}

,let Vi be thesubset of V

(H)

containing vertexi fromeach copy of Fk.Constructfrom Han infinitegraph H(K

)

onthesamevertexsetbyapplyingasubgraph complementationto Vi if andonly if K

(i,

i)

=

1 and byapplying abipartite complementationto a pair Vi,Vj ifandonlyif K

(i,

j)

=

1.Thus, two vertices u

∈

Vi and v

∈

Vj areadjacent in H(K

)

ifandonly ifuv

∈

E

(H)

and K

(i,

j)

=

0 or uv

∈

/

E(H)and K

(i,

j)

=

1. Then,

P

(K

,

Fk

)

isthehereditaryclassconsistingofallthefiniteinducedsubgraphsof H(K

).

Theminimumksuchthat G isk-uniformistheuniformicityofG.Thesecondofthenexttwolemmasfollowsdirectlyfromtheabovedefinitions.

ThefollowingresultwasprovedbyKorpelainenandLozin.Theclassofdisjointunionsofcliquesisacounterexamplefor thereverseimplication.

Lemma8([33]).Anyclassofgraphsofboundeduniformicityiswell-quasi-orderedbythelabelledinducedsubgraphrelation.

Thefollowinglemmafollowsfromthedefinitionofclique-widthandthedefinitionofk-uniformgraphs(seealso [2] for amoregeneralresult).Forcompleteness,weincludeaformalproof.

Lemma9.Foreveryk

≥

1,everyk-uniformgraphhasclique-widthatmost2k.

Proof. Letk

≥

1 andlet H(K

)

be thegraphfromthedefinitionofk-uniformforsome FkandK.Wewillshowthatevery finiteinduced subgraphofH(K

)

hasclique-widthatmost 2k.Let X1

,

X2

,

. . .

be thesets ofverticescorrespondingtoeach

copyofFkintheconstructionofH(K

)

andnotethat

|

Xh|

=

kforallh

≥

1 andthat

|

Xh

∩

Vi|

=

1 forallh

≥

1,i

∈ {

1,

. . . ,

k

}

. Furthermore, iftwo vertices x

∈

Vi and y

∈

Vj do not lie in the same set Xh,then xand y are adjacent ifandonly if K

(i,

j)

=

1.For

≥

0,let H

=

H(K

)[

X1

∪ · · · ∪

X](soH0 istheemptygraph,whichhasnovertices).

[image:7.561.147.392.53.263.2]

Fig. 2.A 3-partite graph partitioned into slicesG0_{, . . . ,G}3_{isomorphic toP} 3.

withthepropertythatoncethisnew2k-expressionhasbeenapplied,foralli

∈ {

1,

. . . ,

k

}

everyvertexinV

(H

+1

)

∩

Vi has labeli.Todothis,wefirsttake the2k-expressionfor H andthen forevery i

∈ {

1,

. . . ,

k

}

,wecreatetheuniquevertexof X+1

∩

Vi withlabelk

+

i andtake thedisjointunionoftheseverticeswiththeconstructed H.Next,fori,j

∈ {

1,

. . . ,

k

}

withi

<

j,if theunique vertexof X+1

∩

Vi isadjacent to theunique vertexof X+1

∩

Vj in H+1,then we apply the

joinoperation

η

k+i,k+j().Next,fori,j

∈ {

1,

. . . ,

k

}

,ifK

(i,

j)

=

1,thenweapplythejoinoperation

η

i,k+j().Finally,forevery i

∈ {1,

. . . ,

k}weapplytherelabeloperation

ρ

k+i→i().Theresulting2k-expressionconstructsH₊1withtherequiredlabels,

so theinductionhypothesis alsoholdsfor

+

1. Byinduction,theinductionhypothesis holds forevery

≥

0 and sowe concludethat H hasclique-widthatmost 2kforevery

.

NowifG isak-uniformgraphthenitisafiniteinducedsubgraphof H(K

),

sothereisan

such thatG isaninduced subgraphofH.Itfollowsthat Ghasclique-widthatmost 2k.Thiscompletestheproof.

2

3. Outlineofourproofmethod

Weproveourresultsonbothwell-quasi-orderabilityandboundednessofclique-widthbyafurthergeneralizationofthe canonicaldecompositionmethodintroducedbyFouquet,GiakoumakisandVanherpe [24] forbipartitegraphs.Asdiscussed inSection1,thismethodwasgeneralizedto3-partitegraphsin [12],butforourpurposesweneedanon-trivialrefinement. WeexplainthisrefinementindetailinSection4,butbelowwesketchthemainideasbehindit.Inordertodothis,wefirst needtointroducesometerminology.

LetG bea3-partitegraphgivenwithapartition ofits vertexsetintothreeindependentsets V1, V2 and V3.Suppose

thateachset Vi canbe partitionedintosets Vi0

,

. . . ,

Vi(the valueof

isthesameforeachi

∈ {

1,2,3

}

) suchthat,taking subscriptsmodulo 3:

– fori

∈ {

1,2,3

}

if j

<

k,thenV_ij iscompletetoVk_i+1andanti-completeto V_ik₊₂.

Fori

∈ {

0,

. . . ,

}

letGi

₌

_G

_[

_Vi

1

∪

V2i

∪

V3i

]

.WesaythatthegraphsGiaretheslicesofG.Iftheslicesbelongtosomegraph

class X,thenwesaythatG canbepartitionedintoslicesfrom X;seeFig.2foranexample. InSection4.1weprovethefollowingtworesults:

(i) every3-partitegraphthatcanbepartitionedintoslicesofclique-widthatmostkhasclique-widthatmost max(3k,6); (ii) foreverysetY of3-partitegraphsthatbelongtosomehereditarygraphclass X:ifthegraphsinY canbepartitioned intoslicesfromsome subclass Z

⊆

X thatiswell-quasi-orderedby thelabelledinduced subgraphrelation,then Y is alsowell-quasi-orderedbythelabelledinducedsubgraphrelation.

withrespecttothepartition

(V

1

,

V2

,

V3

)

ifitcontains norainbow K3 or 3P1 when itsvertexsetispartitionedinthisway.

We saythat G iscuriousifthereisapartition

(V

1

,

V2

,

V3

)

withrespecttowhichitiscurious;note thatbipartite graphs

arecurious,sinceweallowthecasewhenVi

= ∅

forsomei.InSection4.2wewillprovethefollowingresult:

(iii) if theset Xofbipartitegraphs ina hereditarygraphclass X iswell-quasi-orderedbythelabelledinducedsubgraph relationorhaveboundedclique-width,thenthesameistrueforthecuriousgraphsin X.

We will prove result (iii)by showingthat the curiousgraphs in X allow a partition intoslices from X,so that we can indeedapplyresults (i)and (ii).

Finally,inSection 5,weapply result (iii) ontheclassesof

(K

3

,

P1

+

P5

)-free

graphs and

(K

3

,

P2

+

P4

)-free

graphs,in

ordertoprovethattheseclasseshaveboundedclique-widthandarewell-quasi-orderedby thelabelledinducedsubgraph relation.Todothis,weprovetwostructurallemmas.Theselemmasconsiderthecasewhenagraphinoneoftheseclasses contains an induced subgraphisomorphic to C5. Wepartition the remaining vertices ofthe graphinto setsaccording to

their neighbourhood in this C5 and analyse theedges betweenthe vertices inthese sets. We usethis analysisto show

how tousea constantnumberof vertexdeletionsandbipartite complementationsto transformthe graphintoa disjoint unionofcuriousgraphsand3-uniformgraphsintherespectivegraphclass.Thetwolemmasenableustoapplyresult (iii) after combining them with Theorem 3 andLemma 6, respectively, which implythat the bipartite graphs in the classes of

(K

3

,

P1

+

P5

)-free

graphsand

(K

3

,

P2

+

P4

)-free

graphshaveboundedclique-widthandare well-quasi-orderedby the

(labelled)inducedsubgraphrelation,respectively.

4. Partitioning3-partitegraphs

InSection4.1werecallourgraphdecompositionon3-partitegraphs.Wethenshowhowtoextendresultsonbounded clique-width or well-quasi-orderability by the labelled induced subgraph relation from bipartite graphs in an arbitrary hereditaryclass ofgraphs tothe 3-partitegraphsin thisclassthat aredecomposable inthisway.In Section 4.2wegive sufficientconditionsfora3-partitegraphtohavesuchadecomposition.

4.1. Thedecomposition

LetG be a3-partitegraphgivenwithapartitionofitsvertexsetintothreeindependentsets V1, V2 andV3.Suppose

that eachsetVi canbepartitionedintosetsV_i0

,

. . . ,

V_i (thevalueof

isthesameforeachi

∈ {

1,2,3

}

)suchthat,taking subscriptsmodulo 3:

– for i

∈ {

1,2,3

}

if j

<

kthen V_ijiscompleteto V_ik₊₁ andanti-completetoV_ik₊₂.

Fori

∈ {

0,

. . . ,

}

letGi

₌

_G

_[

_Vi

1

∪

V2i

∪

V3i

]

.RecallthatthegraphsGiaretheslicesofG andthatiftheslicesbelongtosome

graphclass X,thenGcanbepartitionedintoslicesfromX.

Lemma 10.If G isa 3-partite graphthatcanbe partitionedintoslicesofclique-widthatmost k,then G hasclique-widthat mostmax(3k,6).

Proof. Since every slice Gj of G hasclique-widthat mostk, itcan be constructedusing ak-expression,which uses the labels 1,

. . . ,

k. Applying relabellingoperations if necessary, we mayassume that at the end of this construction, every vertex haslabel 1. Wecan modify thisk-expression sothat we usethe 3k labels 11

,

. . . ,

k1

,

12

,

. . . ,

k2

,

13

,

. . . ,

k3 instead

(sothemodified expressionwillbe a3k-expression forGj), insuchawaythatatall pointsintheconstruction, foreach i

∈ {

1,2,3

}

everyconstructedvertexinVi hasalabelin

{

1i,

. . . ,

ki}.Todothiswereplace:

– creationoperationsi(v)byij(v

)

ifv

∈

Vj,

– relabeloperations

ρ

j→k()by

ρ

j1→k1

(

ρ

j2→k2

(

ρ

j3→k3

()))

and – joinoperations

η

j,k()by

η

j1,k1

(

η

j1,k2

(

η

j1,k3

(

η

j2,k1

(

η

j2,k2

(

η

j2,k3

(

η

j3,k1

(

η

j3,k2

(

η

j3,k3

())))))))).

This new3k-expressionuses 3k labelsand atthe endofit, every vertexin Vi is labelledwith label 1i.We find such a 3k-expressionEj _{foreverysliceG}j _{ofG independently.}

We now prove by induction on j

≥

0 that there is a max(3k,6)-expression which constructs G

[

V

(G

0

₎

_{∪ · · · ∪}

_V

_(G

j

_)]

in such a waythat every vertex in Vi is labelled withlabel 2i. For j

=

0 we simply take the 3k-expression E0 for G0 andthenapplythreerelabellingoperations

ρ

11→21,

ρ

12→22 and

ρ

13→23.Supposetheinductionhypothesisholdsforsome j

≥

0 and let Fj _{be the corresponding max(3k,6)-expression for G}

_[

_V

_(G

0

₎

_{∪ · · · ∪}

_V

_(G

j

_)]

_{. We take thedisjoint union of} the max(3k,6)-expressions Fj _{and E}j+1_{. Next, we apply join operations}

_η

relabelling operations

ρ

11→21,

ρ

12→22 and

ρ

13→23. This yields a max(3k,6)-expression for G

[

V

(G

0

₎

_{∪ · · · ∪}

_V

_(G

j+1

₎

_]

_in

such awaythatevery vertexinVi islabelledwith 2i.(Note thatintheobtainedexpression weonlyeverusethelabels

{

11

,

. . . ,

max(k,2)1

,

12

,

. . . ,

max(k,2)2

,

13

,

. . . ,

max(k,2)3

}

,soitisindeedamax(3k,6)-expression.)Thereforetheinduction

hypothesisholdsforevery j

≥

0.Byinduction,itfollowsthatG hasclique-widthatmost max(3k,6).

2

Lemma11.Let X beahereditarygraphclasscontainingaclass Z .Let Y bethesetof3-partitegraphsin X thatcanbepartitionedinto slicesfrom Z .If Z iswell-quasi-orderedbythelabelledinducedsubgraphrelation,thensois Y .

Proof. Foreach graphG inY,wemayfixapartition intoindependentsets

(V

1

,

V2

,

V3

)

withrespectto whichthegraph

canbepartitionedintoslicesfromZ.Let

(L

1

,

≤

1

)

bethewell-quasi-orderwithL1

= {

1,2,3

}

inwhicheverypairofdistinct

elementsisincomparable.ByLemma7,we needonlyconsiderlabellingsofgraphsinY oftheform

(i,

l(v))wherev

∈

Vi and l(v) belongs to an arbitrarywell-quasi-order L. Suppose G can be partitioned into slices G0

_,

_{. . . ,}

_Gk_{, withvertices} labelledasin G.Notethat the orderedlist G0

_,

_{. . . ,}

_Gk _{completelyspecifies theedges of G andthe labelson its vertices.} Indeed, two vertices x,y

∈

V

(G

i

)

are adjacent in G if and only if they are adjacent in Gi and two vertices x

∈

V

(G

i

),

y

∈

V

(G

j

)

withi

<

jareadjacentinGifandonlyifx

∈

Vkandy

∈

Vk+1 forsomek

∈ {

1,2,3

}

(subscriptstakenmodulo 3).

Now consider two labelled graphs G,H

∈

Y with corresponding partitions into slices G0

_,

_{. . . ,}

_Gk _{and H}0

_,

_{. . . ,}

_{H. If}

(H

0

_,

_{. . . ,}

_H

₎

_{is smallerthan orequal to}

_(G

0

_,

_{. . . ,}

_Gk

₎

_{underthe subsequence relation,then H is a labelledinduced} sub-graphofG.TheresultfollowsbyLemma3.

2

4.2. Curiousgraphs

LetG bea3-partitegraphgiventogetherwithapartitionofitsvertexsetintothreeindependentsets V1, V2 and V3.

Recallthataninduced K3 or 3P1 inG israinbow ifit hasexactlyone vertexineach set Vi.Alsorecallthat G iscurious

withrespecttothepartition

(V

1

,

V2

,

V3

)

ifitcontainsnorainbow K3 or 3P1 whenitsvertexsetispartitionedinthisway,

andthat G iscuriousifthere isapartition

(V

1

,

V2

,

V3

)

withrespecttowhichit iscurious.Inthissection wewillprove

thatgivenahereditaryclass X,ifthebipartitegraphsinXarewell-quasi-orderedbythelabelledinducedsubgraphrelation orhaveboundedclique-width,thenthesameistrueforthecuriousgraphsin X.

Alinearorder

(x

1

,

x2

,

. . . ,

xk)oftheverticesofanindependentsetI is

– increasingifi

<

jimpliesN(xi)

⊆

N(xj), – decreasingifi

<

jimpliesN(xi)

⊇

N(xj), – monotoneifitiseitherincreasingordecreasing.

Bipartite graphs that are 2P2-free are also known asbipartite chain graphs. It is well known (and easy to verify) that

a bipartite graph G is 2P2-free ifand onlyif the verticesin each independent set ofthe bipartition admit a monotone

ordering. Suppose G is a curious graph with respect to some partition

(V

1

,

V2

,

V3

).

We say that (with respect to this

partition)thegraphG isacuriousgraphoftype tifexactlytofthegraphsG

[

V1

∪

V2

]

,G

[

V1

∪

V3

]

andG

[

V2

∪

V3

]

contain

aninduced 2P2.

4.2.1. Curiousgraphsoftype0and1

NotethatifGisacuriousgraphoftype 0 or 1 withrespecttothepartition

(V

1

,

V2

,

V3

)

thenwithoutlossofgenerality,

wemayassumethatG

[

V1

∪

V2

]

andG

[

V1

∪

V3

]

areboth2P2-free.

Lemma12.Let G beacuriousgraphwithrespectto

(V

1

,

V2

,

V3

),

suchthatG

[

V1

∪

V2

]

andG

[

V1

∪

V3

]

areboth2P2-free.Thenthe

verticesof V1admitalinearorderingwhichisdecreasinginG

[

V1

∪

V2

]

andincreasinginG

[

V1

∪

V3

]

.

Proof. ForasetS

⊆

V,weuseNS

(u)

:=

N(u)

∩

Stodenotethesetofverticesin Sthatareadjacenttou.Wemaychoose a linear order x1

,

. . . ,

x of the vertices of V1 according to their neighbourhood in V2, breaking ties according to their

neighbourhoodinV3i.e.anordersuchthat:

(i) ifNV2

(x

i)

NV2

(x

j)theni

<

j and

(ii) ifNV₂

(x

i)

=

NV₂

(x

j)andNV3

(x

i)

NV3

(x

j)theni

<

j.

ClearlysuchanorderingisdecreasinginG

[

V1

∪

V2

]

.

Suppose,forcontradiction, thatthisorderis notincreasing inG

[

V1

∪

V3

]

.Then theremustbe indicesi

<

j such that

NV₃

(x

i)

NV3

(x

j).Then NV2

(x

i)

=

NV2

(x

j) by Property (ii).ByProperty (i) itfollowsthat NV2

(x

i)

NV2

(x

j).Thismeans that there are vertices y

∈

NV2

(xi)

\

NV2

(xj)

and z

∈

NV3

(xi)

\

NV3

(xj).

Now if y is adjacent to z then G

[

xi,y,z

]

is a rainbow K3 and if y isnon-adjacent to z then G

[

xj,y,z

]

is arainbow 3P1.This contradictionimpliesthat the orderis

Lemma13.If G isacuriousgraphoftype0or1withrespecttoapartition

(V

1

,

V2

,

V3

)

then G canbepartitionedintoslicesthatare

bipartite.

Proof. SinceGisacuriousgraphoftype 0 or 1,withoutlossofgenerality,wemayassumethatG

[

V1

∪

V2

]

andG

[

V1

∪

V3

]

are both 2P2-free. Let

= |

V1

|

(wewill partition G into

+

1 slices G0

,

. . . ,

G). Letx1

,

. . . ,

x be alinear order on V1

satisfyingLemma12.LetV₁0

= ∅

andfori

∈ {

1,

. . . ,

}

,let V₁i

= {

xi}.Wepartition V2 and V3 asfollows.Fori

∈ {

0,

. . . ,

}

,

let Vi

2

= {

y

∈

V2

|

xjy

∈

E

(G)

if and only if j

≤

i

}

. For i

∈ {

0,

. . . ,

}

, let V3i

= {

z

∈

V3

|

xjz

∈

/

E

(G)

if and only if j

≤

i

}

. In particular,notethattheverticesofV

2

∪

V30andV20

∪

V3arecompleteandanti-completeto V1,respectively.Thefollowing

propertieshold:

– If j

<

k,then V₁jiscompleteto Vk

2 andanti-completeto V3k.

– If j

>

k,then V₁jisanti-completetoVk

2 andcompleteto V3k.

If j

<

k and y

∈

V₂j is non-adjacentto z

∈

V₃k then G

[

xk,y,z

]

isa rainbow 3P1,a contradiction. If j

>

k and y

∈

V2j is

adjacenttoz

∈

Vk₃ thenG

[

xj,y,z

]

isarainbow K3,acontradiction.Itfollowsthat:

– If j

<

k,then V₂jiscompleteto Vk

3.

– If j

>

k,then V₂jisanti-completetoVk₃.

Fori

∈ {

0,

. . . ,

}

,letGi

=

G

[

V₁i

∪

V₂i

∪

V₃i

]

.TheabovepropertiesabouttheedgesbetweenthesetsVi_jimplythefollowing statements:

– If j

<

k,then V₁jiscompleteto V₂kandanti-completeto V₃k. – If j

<

k,then V₂jiscompleteto V₃kandanti-completeto V₁k. – If j

<

k,then V₃jiscompleteto Vk

1 andanti-completeto V2k.

ThusGcanbepartitionedintotheslicesG0

,

. . . ,

G_{.Now,foreachi}

_{∈ {}

_0,

_{. . . ,}

_}

_,Vi

1isanti-completetoV3i,soeverysliceG

i

isbipartite.Thiscompletestheproof.

2

4.2.2. Curiousgraphsoftype2and3

Lemma14.Fixt

∈ {

2,3

}

.If G isacuriousgraphoftype t withrespecttoapartition

(V

1

,

V2

,

V3

)

then G canbepartitionedintoslices

oftypeatmostt

−

1.

Proof. Fixt

∈ {

2,3

}

andlet G bea curiousgraphoftype t withrespecttoa partition

(V

1

,

V2

,

V3

).

Wemayassumethat

G

[

V1

∪

V2

]

containsaninduced 2P2.

Claim1.Givena2P2inG

[

V1

∪

V2

]

,everyvertexof V3hasexactlytwoneighboursinthe2P2andtheseneighbourseitherbothlie

in V1orbothliein V2.

Letx1

,

x2

∈

V1andy1

,

y2

∈

V2inducea 2P2inGsuchthatx1isadjacenttoy1butnotto y2andx2isadjacentto y2but

notto y1.Consider z

∈

V3.Fori

∈ {

1,2

}

,ifz isadjacenttobothxi andyi,thenG

[

xi,yi,z

]

isarainbow K3,acontradiction.

Therefore z can haveatmostone neighbourin

{

x1

,

y1

}

andatmostone neighbourin

{

x2

,

y2

}

.If z isnon-adjacentto x1

and y2 thenG

[

x1

,

y2

,

z

]

isarainbow 3P1,acontradiction.Therefore,zcanhaveatmostonenon-neighbourin

{

x1

,

y2

}

,and

similarly z canhave atmostone non-neighbourin

{

x2

,

y1

}

.Therefore,ifz isadjacentto x1,thenitmustbe non-adjacent

to y1,soitmustbeadjacenttox2,soitmustbenon-adjacentto y2.Similarly,ifz isnon-adjacenttox1 then itmustbe

adjacentto y2,soitmustbenon-adjacenttox2,soitmustbeadjacentto y1.HenceClaim1follows.

Consideramaximalset

{

H1

_,

_{. . . ,}

_Hq

_}

_{ofvertex-disjointsetsthatinducecopiesof 2P}

2inG

[

V1

∪

V2

]

.Wesaythatavertex

ofV3distinguishestwographsG

[

Hi

]

andG

[

Hj

]

ifitsneighboursinHiandHj donotbelongtothesamesetVk.Wegroup thesesets Hi intoblocks B1

,

. . . ,

Bp that arenot distinguished by anyvertexof V3.Inother words,every Bi contains at

leastone 2P2andeveryvertexofV3 iscompletetooneofthesets Bi

∩

V1 andBi

∩

V2andanti-completetotheother.For

j

∈ {

1,2

}

,letBi

j

=

Bi

∩

Vj.Wedefinearelation

<B

ontheblocksasfollows:

– Bi

<B

Bj holdsifBi₁ iscompletetoB₂j,while B₂i isanti-completeto B₁j.

[image:11.561.154.387.51.254.2]

Fig. 3.Two blocksBi_andBj_withBi_<BBj_{and a vertexz∈V}

3distinguishing them.

Claim2.Let Biand Bjbedistinctblocks.Thereisavertexz

∈

V3 thatdistinguishes Bi and Bj.If z iscompleteto Bi2

∪

B

j

1and

anti-completetoBi

1

∪

B

j

2thenBi

<B

Bj(seealsoFig.3).If z iscompletetoB

j

2

∪

Bi1andanti-completeto B

j

1

∪

Bi2thenBj

<B

Bi.

Furthermore,oneofthesecasesmustapply,whichimpliesthateitherBi

<B

BjorBj

<B

Biforeverypairofdistinctblocks Biand Bj.

Bydefinitionoftheblocks Bi _andBj _{theremustbeavertexz}

_∈

_V

3 thatdistinguishesthem.Withoutlossofgenerality

we mayassume that z iscompleteto B₂i

∪

B₁j andanti-completeto B₁i

∪

B₂j.Itremains toshow that Bi

_<B

_Bj_{.If y}

_∈

_Bi

2

isadjacentto z

∈

B₁j thenG

[

x,y,z

]

isarainbow K3,acontradiction.If y

∈

B1i isnon-adjacenttoz

∈

B

j

2 then G

[

x,y,z

]

is

a rainbow 3P1,a contradiction. Therefore Bi2 is anti-complete to B

j

1 and Bi1 is completeto B

j

2.It follows that Bi

<B

Bj.

Claim2followsbysymmetry.

Claim3.Therelation

<B

istransitive.

SupposethatBi

_<B

_Bj_andBj

_<B

_Bk_.SinceBj_andBk_{aredistinct,theremustbeavertexz}

_∈

_V

3 thatdistinguishesthem,

andby combiningClaim2withthefact that Bj

_<B

_Bk _{it followsthat z must becomplete to B}j

2

∪

Bk1 andanti-complete

toB₂j

∪

Bk₁.Supposex

∈

Bi₁and y

∈

B₂j.Then xisadjacentto y,since Bi

<B

Bjandy isadjacenttozbychoiceofz.Now x ismustbe non-adjacentto z otherwise G

[

x,y,z

]

wouldbea rainbow K3.Therefore z isanti-complete to Bi1.Recallthat

everyvertexofV3 iscompletetooneofBi1andBi2andanti-completetotheother(duetoClaim1).Thereforeziscomplete

to Bi

2,andsozdistinguishesBi andBk.ByClaim2itfollowsthatBi

<B

Bk.ThiscompletestheproofofClaim3.

CombiningClaims1–3,wefindthat

<B

isalinearorderontheblocks.Weobtainthefollowingconclusion,whichwe callthechainproperty.

Claim4.ThesetofblocksadmitsalinearorderB1

<B

B2

<B

· · ·

<B

Bpsuchthat

(i) if i

<

j then Bi

1iscompleteto B

j

2,while Bi2isanti-completeto B

j

1and

(ii) foreachz

∈

V3thereexistsani

∈ {

0,

. . . ,

p

}

suchthatifj

≤

i then z iscompleteto B₂jandanti-completeto B₁jandifj

>

i then z

isanti-completeto B₂jandcompleteto B₁j.

NextconsiderthesetofverticesinV1

∪

V2thatdonotbelongtoanysetBi.LetRdenotethissetandnotethat G

[

R

]

is

2P2-freebymaximalityoftheset

{

H1

,

. . . ,

Hq

}

.Fori

∈ {

1,2

}

letRi

=

R

∩

Vi.

Claim5.Ifx

∈

R1hasaneighbourin Bi₂,then x iscompleteto B₂i+1,andif x hasanon-neighbourin Bi₂,then x isanti-completeto Bi₂−1.

Ifx

∈

R2hasanon-neighbourinBi1,thenx isanti-completeto B

i+1

1 ,andif x hasaneighbourin B1i,then x iscompleteto B

i−1 1 .

Suppose,forcontradiction,thatx

∈

R1 isadjacentto y

∈

B2i andnon-adjacentto y

∈

B

i+1

2 .Consideravertexz

∈

V3that

distinguishesBi_andBi+1_.SinceBi

_<B

_Bi+1_{,Claim2impliesthatziscompletetoB}i+1 1

∪

B

i

2andanti-completetoB

i+1 2

∪

B

i