Contraction
and
deletion
blockers
for
perfect
graphs
and H-free
graphs
✩
Öznur
Ya ¸sar Diner
a,
1,
Daniël Paulusma
b,
2,
Christophe Picouleau
c,
Bernard Ries
d,
∗
aKadir Has University, Istanbul, Turkey bDurham University, Durham, UK cCNAM, Laboratoire CEDRIC, Paris, France dUniversity of Fribourg, Fribourg, Switzerland
a
b
s
t
r
a
c
t
We study the following problem: for given integers d, k and graph G, can we reduce some fixed graph parameter
π
of G by at least d viaat most
k graphoperations from
some fixed set S?As parameters we take the chromatic number
χ
, clique numberω
and independence numberα
, and as operations we choose edge contraction ec and vertex deletion vd. We determine the complexity of this problem for S= {ec} and S= {vd} andπ
∈ {χ
,ω
,α
}for a number of subclasses of perfect graphs. We use these results to determine the complexity of the problem for S= {ec} and S= {vd} andπ
∈ {χ
,ω
,α
}restricted to H-free
graphs.
1. Introduction
AtypicalgraphmodificationproblemaimstomodifyagraphG,viaasmallnumberofoperationsfromaspecifiedset S, intosome othergraph H thathasacertain desiredproperty,whichusually describesa certaingraphclass
G
towhich Hmustbelong.Inthiswayavarietyofclassicalgraph-theoreticproblemsiscaptured.Forinstance,ifonlyk vertexdeletions areallowedandH mustbeanindependentsetoraclique,weobtainthe IndependentSetor Clique problem,respectively. Now,insteadoffixing aparticular graphclass
G
,wefixacertaingraphparameterπ
.Thatis,forafixed set S of graph operations, we ask, givena graph G,integers k and d,whether G canbe transformed intoa graph G by using atmostk operationsfrom S, such that
π
(
G)
≤
π
(
G)
−
d.The integer d is calledthethreshold. Such problemsare calledblocker problems,asthesetofverticesoredgesinvolved“block”somedesirablegraphproperty,suchasbeingcolourablewithonly✩ AnumberofresultsinthispaperhaveappearedinextendedabstractsoftheproceedingsofCIAC 2016 [16],ISCO2016 [36],LAGOS2017 [35] and
TAMC2017 [37].
*
Correspondingauthor.E-mail addresses:oznur.yasar@khas.edu.tr(Ö.Y. Diner),daniel.paulusma@durham.ac.uk(D. Paulusma),christophe.picouleau@cnam.fr(C. Picouleau),
bernard.ries@unifr.ch(B. Ries).
1 AuthorsupportedbyMarieCurieInternationalReintegrationGrantPIRG07/GA/2010/268322. 2 AuthorsupportedbyEPSRC(EP/K025090/1)andtheLeverhulmeTrust(RPG-2016-258).
Published in "Theoretical Computer Science 746 (2018) 49-72"
which should be cited to refer to this work.
afewcolours.Identifyingthepartofthegraphresponsibleforasignificantdecreaseoftheparameterunderconsideration givescrucialinformationonthegraph.
Blockerproblemshavebeengivenmuch attentionover thelastfew years[2–4,13,38,39,42,44].Graphparameters con-sideredwerethechromatic number,theindependencenumber,thecliquenumber,thematchingnumber,theweightofa minimumdominatingsetandthevertexcovernumber.Sofar,thesetS alwaysconsistedofasinglegraphoperation,which wasavertexdeletion,anedgedeletionoranedgeaddition.Inthispaper,wekeeptherestrictiononthesizeofS byletting
S consistofeitherasinglevertexdeletionor,forthefirsttime,asingleedgecontraction.Asgraphparameters,weconsider theindependencenumber
α
,thecliquenumberω
andthechromaticnumberχ
.Beforewecandefineourproblemsformally,wefirstneedtogivesometerminology.Thecontraction ofanedgeuv ofa graphG removestheverticesu and v fromG,andreplacesthembyanewvertexmadeadjacenttopreciselythosevertices that were adjacenttou or v in G (neitherintroducingself-loopsnormultipleedges).Wesay that G canbe k-contracted
ork-vertex-deleted into agraph G, ifG canbe modified into G bya sequence ofatmost k edgecontractions orvertex deletions,respectively.Welet
π
denotethe(fixed)graphparameter;asmentioned,inthispaperπ
belongsto{
α
,
ω
,
χ
}
.Wearenowreadytodefineourdecisionproblemsinageneralway: ContractionBlocker(
π
)Instance: a graph G and two integers d
,
k≥
0Question: can G be k-contracted into a graph Gsuch that
π
(
G)
≤
π
(
G)
−
d?DeletionBlocker(
π
)Instance: a graph G and two integers d
,
k≥
0Question: can G be k-vertex-deleted into a graph Gsuch that
π
(
G)
≤
π
(
G)
−
d?If we remove d from the input and fix it instead, then we call the resulting problems d-ContractionBlocker(
π
) andd-DeletionBlocker(
π
),respectively.d-ContractionBlocker(
π
)Instance: a graph G and an integer k
≥
0Question: can G be k-contracted into a graph Gsuch that
π
(
G)
≤
π
(
G)
−
d?d-DeletionBlocker(
π
)Instance: a graph G and an integer k
≥
0Question: can G be k-vertex-deleted into a graph Gsuch that
π
(
G)
≤
π
(
G)
−
d?The goal of our paper is to increase our understanding of the complexities of ContractionBlocker(
π
) and Dele-tionBlocker(π
)forπ
∈ {
ω
,
χ
,
α
}.
Inorder todoso, wewill alsoconsiderthe problems d-ContractionBlocker(π
)andd-deletionblocker(
π
).1.1. Knownresultsandrelationstootherproblems
Itisknownthat DeletionBlocker(
α
)ispolynomial-timesolvableforbipartitegraphs,asprovenbothbyBazgan, Touba-line and Tuza [3] andCosta, de Werra and Picouleau [13]. The former authors alsoproved that DeletionBlocker(α
) is polynomial-timesolvableforcographsandgraphsofboundedtreewidth.Thelatterauthorsalsoprovedthatforπ
∈ {
ω
,
χ
}
, DeletionBlocker(π
) is polynomial-time solvable for cobipartite graphs. Moreover, they showed that forπ
∈ {
ω
,
χ
,
α
}
, DeletionBlocker(π
)is NP-completefortheclassofsplitgraphs,butbecomespolynomial-timesolvableforthisgraphclass ifd isfixed.Byusinganumberofexampleproblemswewillnowillustratehowtheblockerproblemsstudiedinthispaperrelateto a numberofotherproblemsknownintheliterature.Aswe willsee,thisimmediatelyleadstonewcomplexityresultsfor theblockerproblems.
1. HadwidgerNumberandClubContraction. The ContractionBlocker(
α
) problemgeneralizesthewell-known Hadwiger Number problem, which isthat oftesting whethera graphcan be contractedinto thecomplete graph Kr on r vertices forsome givenintegerr. Indeed,weobtainthelatterproblemfromthefirstby restrictinginstancestoinstances(
G,
d,
k)
whered
=
α
(
G)
−
1 andk= |
V(
G)
|
−
r.NotethatthediameterandindependencenumberofKr arebothequalto 1.Hence,one canalsogeneralize HadwigerNumberinanotherway:the ClubContractionproblem(seee.g. [21])isthatoftesting whether agraph G canbe k-contracted intoa graph withdiameteratmost s for some givenintegers k and s. As such, ContractionBlocker(
α
)canbeseenasanaturalcounterpartof ClubContraction.2. Graphtransversals. Blocker problems generalize the so-called graph transversal problems. To explain the latter type of problems,for a family of graphs
H
, theH
-transversal problem isto test ifa graph G can be k-vertex-deleted, for some integer k, into a graph G that has no induced subgraph isomorphic to a graph inH
. For instance, the problem{
K2}
-transversalisthesameas VertexCover.Herearesomeexamplesofspecificconnectionsbetweengraphtransversals andblockerproblems.– Let
H
be thefamily{
Kp|
p≥
2}
ofall completegraphsonatleasttwovertices.ThenH
-transversalisequivalentto DeletionBlocker(ω
)restrictedtoinstances(
G,
d,
k)
withd=
ω
(
G)
−
1.– InourpaperwewillprovethatforagraphG withatleastoneedgeandaninteger k
≥
1,theinstance(
G,
ω
(
G)
−
1,
k)
isayes-instanceof DeletionBlocker(
ω
)ifandonlyif(
G,
k)
isayes-instanceof VertexCover.– The OddCycleTransversalproblemis totest whethera givengraph canbe made bipartite byremoving atmostk verticesforsome givenintegerk
≥
0.Thisproblemis NP-complete [31],anditis equivalentto DeletionBlocker(χ
) forinstances(
G,
d,
k)
whered=
χ
(
G)
−
2.– The d-Transversal or d-Cover problem [13] istodecidewhetheragraphG
= (
V,
E)
containsaset V thatintersects eachmaximumsetsatisfyingsomespecifiedpropertyπ
byatleastd vertices.Forinstance,ifthepropertyisbeingan independentset, 1-Transversal isequivalentto 1-DeletionBlocker(α
).3. BipartiteContraction. Theproblem BipartiteContractionistotestwhetheragraphcan bemadebipartite byatmost
k edgecontractions.Heggerneset al. [26] provedthatthisproblemis NP-complete. Itisreadilyseenthat 1-Contraction Blocker(
χ
)and BipartiteContractionareequivalentforgraphsofchromaticnumber 3.4.Maximuminducedbipartitesubgraphs. The MaximumInducedBipartiteSubgraphproblemistodecideifagivengraph containsaninducedbipartitesubgraphwithatleastk verticesforsomeinteger k.Addario-Berryet al.[1] provedthatthis problemis NP-complete fortheclassof3-colourableperfectgraphs.Weobservethat, for3-colourablegraphs, 1-Deletion Blocker(
χ
)isequivalentto MaximumInducedBipartiteSubgraph.5. Cores. Thetwoproblems 1-DeletionBlocker(
α
)and 1-DeletionBlocker(ω
)areequivalenttotestingwhethertheinput graph contains a set of S of size at most k that intersects every maximum independent set orevery maximumclique, respectively.Ifk=
1,thesetwoproblemsbecomeequivalenttotestingwhethertheinputgraphcontainsavertexthatisin everymaximumindependentset,orineverymaximumclique,respectively.Inparticular,theintersectionofall maximum independent sets isknown asthe core of a graph.Properties ofthe corehave been well studied (see,for example, [25,29,30]). Inparticular, Boros,Golumbic andLevit [7] proved that determining ifthecoreof agraph hassize atleast
is co-NP-hardforeveryfixed
≥
1.Taking=
1 givesco-NP-hardnessof 1-DeletionBlocker(α
).6. Criticalverticesandedges. Therestrictiond
=
k=
1 hasalsobeenstudiedwhenπ
=
χ
.Avertexofagraph G iscriticalifits deletionreducesthechromatic numberof G by 1. Anedge ofagraphis critical orcontraction-critical if itsdeletion or contraction, respectively, reducesthe chromatic number of G by 1. The problems CriticalVertex, CriticalEdge and Contraction-Criticalaretotest ifagraphhasacriticalvertex, criticaledge orcontraction-criticaledge,respectively.We note that the first two problems are the restrictions of ContractionBlocker(
χ
) and DeletionBlocker(χ
) to instances(
G,
d,
k)
whered=
k=
1.Complexitydichotomiesexistforeachofthethreeproblemson H -freegraphs,andmoreoverthe lattertwo problemsareshownto beequivalent [35]. Graphswithacritical(or equivalentlycontraction-critical) edgeare alsocalledcolour-critical (see,forinstance, [40]).Duetolinkstoproblemsastheonesabove,itisofnosurprisethatmanyresultsforblockerproblemsareknownimplicitly intheliteraturealreadyinvarioussettings.Forexample,Belmonteetal. [5] provedthat1-ContractionBlocker(
),where
denotesthe maximumvertex-degree,is NP-completeevenforsplitgraphs.We makeuseofseveralknowncomplexity resultsforsomeoftherelatedproblemsstatedaboveforprovingourresults.
1.2. Ourresults
InSection 1.1we mentioned that DeletionBlocker(
π
) is knownto be NP-completeforπ
∈ {
α
,
ω
,
χ
}
even when re-strictedtospecialgraphclasses.Non-surprisingly, ContractionBlocker(π
)is NP-complete forπ
∈ {
α
,
ω
,
χ
}
aswell(this followsfromourresultsinSection8,butitisalsoeasytoshowthisdirectly).Dueto theabove, itisnaturalto restrictinputsto some specialgraphclassesinorderto obtaintractable resultsand to increase our understanding ofthe computational hardness of the problems.Note that it is not always clear whether ContractionBlocker(
π
)and DeletionBlocker(π
)belong to NP whenrestrictedtoagraphclassG
.However,whenG
is closedunderedgecontractionorvertexdeletion,respectively,andπ
canbeverifiedinpolynomialtime,thenmembership of NP holds:wecantakeascertificatethesequenceofedgecontractionsorvertexdeletions,respectively.Wepresentourresultsintwoparts.
PartI. Inthe first partofour paperwe focus onthe class ofperfectgraphs anda numberof well-known subclassesof perfectgraphs.Mostoftheseclassesarenotonlyclosedundervertexdeletionbutalsounderedgecontraction.Thisenables ustogetunifiedresultsforthecases
π
=
ω
andπ
=
χ
(notethatω
=
χ
holdsbydefinitionofaperfectgraph).AnotherTable 1
Summaryofresultsforsubclassesofperfectgraphs.Here NP-cand NP-hstandfor NP-completeand NP-hard, respec-tively,whereas P standsforpolynomial-timesolvable.Ablankentryindicatesanopencase.Allentriesapartfromthe fivereferencedonesandtheirconsequencesforchordalandperfectgraphsarenewresultsproveninPart Iofthispaper.
Class Contraction Blocker(π) Deletion Blocker(π)
π=α π=ω=χ π=α π=ω=χ
Tree P P P[3,13] P
Bipartite (χ=2) NP-h P P[3,13] P Cobipartite d=1: NP-c NP-c; d fixed: P P P[13]
Cograph P P P[3] P
Split NP-c; d fixed: P NP-c; d fixed: P NP-c; d fixed: P [13] NP-c; d fixed: P [13]
Interval P P
Chordal NP-c d=1: NP-c NP-c d=1: NP-c
C4-free perfect &ω=3 d=1: NP-c
Perfect d=1: NP-h d=1: NP-h NP-c d=1: NP-c
reasonforconsideringsubclassesofperfectgraphsisthat
α
,ω
,χ
canbecomputedinpolynomialtimeforperfectgraphs; Grötschel, Lovász, andSchrijver [23] proved thisforχ
andthusforω
,whereas theresultforα
followsfromcombining thisresultwiththefactthatperfectgraphsareclosedundercomplementation(thelatterobservationfollowsdirectlyfrom the Strong PerfectGraph Theorem [9];see alsoTheorem 3). Thishelpsuswith findingtractable resultsorat leastwith obtaining membershipof NP (if in additionthe subclassunder consideration isclosed underedge contraction orvertex deletion).Table 1givesanoverviewoftheknownresultsandournewresultsfortheclassesofperfectgraphs weconsider. We have unified resultsfor thecases
π
=
ω
andπ
=
χ
even forthe perfectgraphclasses inthis table that are not closed underedgecontraction,namelytheclassesofbipartitegraphs;C4-freeperfectgraphswithcliquenumber 3;andtheclassofperfectgraphsitself.Astheclassofperfectgraphsisnotclosedunderedgecontractionwecouldforperfectgraphsonly deducethatthethreecontractionblockerproblemsare NP-hard(evenifd
=
1).Astheclassofcographscoincideswiththe classofP4-freegraphs(where Prdenotesther-vertexpath)andsplitgraphsare P5-free,thecorrespondingrowsinTable1show a complexity jumpofall ourproblemsfor Pt-free graphsfromt
=
4 tot=
5. Recallalso fromSection 1.1that theHadwigerNumberproblemisaspecialcaseof ContractionBlocker(
α
). Assuch,ourpolynomial-timeresultinTable1for ContractionBlocker(α
)restrictedtocographsgeneralizesaresultofGolovachetal. [21],whoprovedthatthe Hadwiger Numberproblemispolynomialtimesolvableoncographs.PartII. Inthesecondpartofourpaperwegiveseveraldichotomyresults.Firstwegive,for
π
∈ {
α
,
ω
,
χ
}
,complete classi-ficationsof DeletionBlocker(π
)and ContractionBlocker(π
)dependingonthesizeofπ
,thatis,weprovethefollowing theorem.Theorem1.Thefollowingsixdichotomieshold:
(i) ContractionBlocker(
α
)ispolynomial-timesolvableforgraphswithα
=
1 and 1-ContractionBlocker(α
)is NP-completeforgraphswith
α
=
2;(ii) ContractionBlocker(
χ
)ispolynomial-timesolvableforgraphswithχ
=
2 and 1-ContractionBlocker(χ
)is NP-completeforgraphswith
χ
=
3;(iii) ContractionBlocker(
ω
)ispolynomial-timesolvableforgraphswithω
=
2 and 1-ContractionBlocker(ω
)is NP-completeforgraphswith
ω
=
3;(iv) DeletionBlocker(
α
)ispolynomial-timesolvableforgraphswithα
=
1 and 1-DeletionBlocker(α
)is NP-completeforgraphswith
α
=
2;(v) DeletionBlocker(
χ
)ispolynomial-timesolvableforgraphswithχ
=
2 and 1-DeletionBlocker(χ
)is NP-completeforgraphswith
χ
=
3;(vi) DeletionBlocker(
ω
)ispolynomial-timesolvableforgraphswithω
=
1 and 1-DeletionBlocker(ω
)is NP-completeforgraphswith
ω
=
2.Inparticularweextendthehardnessproof ofTheorem1(iii)inordertoobtainthehardness resultforC4-freeperfect
graphs with
ω
=
3 inTable 1. We note that some of the resultsin Table 1,such as thisresult, may atfirst sight seem somewhat arbitrary.However, we needthe result forC4-free perfect graphs withω
=
3 and other results ofTable 1 toprove ourother resultsofthesecond partofourpaper.Namely,bycombiningtheresultsforsubclassesofperfectgraphs with other results,we obtain complexity dichotomies forour six blockersproblems restricted to H -free graphs, that is, graphs that donot contain some (fixed)graph H as an inducedsubgraph. Thesedichotomies are statedinthe following summary; here, Pr is ther-vertex path, C3 is thetriangle, andthepaw is the trianglewithan extra vertexadjacent to
exactlyone vertexofthetriangle,whereas
⊆
i denotestheinducedsubgraphrelationand⊕
denotesthedisjointunionofTheorem2.LetH beagraph.Thenthefollowingholds:
(i) IfH
⊆
iP4,then DeletionBlocker(α
)ispolynomial-timesolvableforH -freegraphs,otherwiseitis NP-hardorco-NP-hardforH -freegraphs.
(ii) If H
⊆
i P4,then ContractionBlocker(α
)ispolynomial-timesolvableforH -freegraphs,otherwiseitis NP-hardforH -freegraphs.
(iii) IfH
⊆
iP4,then DeletionBlocker(ω
)ispolynomial-timesolvableforH -freegraphs;otherwiseitis NP-hardorco-NP-hardforH -freegraphs.
(iv) LetH
=
C3⊕
P1.IfH⊆
iP4orH⊆
ipaw,then ContractionBlocker(
ω
)
ispolynomial-timesolvableforH -freegraphs,other-wiseitis NP-hardorco-NP-hardforH -freegraphs.
(v) IfH
⊆
iP1⊕
P3orH⊆
iP4,then DeletionBlocker(
χ
)
ispolynomial-timesolvableforH -freegraphs,otherwiseitis NP-hardorco-NP-hardforH -freegraphs.
(vi) IfH
⊆
iP4,then ContractionBlocker(
χ
)
ispolynomial-timesolvableforH -freegraphs,otherwiseitis NP-hardorco-NP-hardforH -freegraphs.
Statements(i), (ii), (iii), (v), (vi) of Theorem2 correspond to complete complexity dichotomies, whereas there is one missingcaseinstatement (iv).Inparticularwenotethatstatements (v)and (vi)donotcoincidefordisconnectedgraphs H . WealsoobservefromTheorem2(i)that DeletionBlocker(
α
)iscomputationallyhardfortriangle-freegraphs;infactwe willshow co-NP-hardness evenifd=
k=
1.Thisincontrastto theproblembeingpolynomial-timesolvableforbipartite graphs,asshownin [3,13] (seealsoTable1).1.3. Paperorganization
Section2containsnotationandterminology.
Sections3–7containtheresultsmentionedinPart I.Tobe moreprecise,Section3contains ourresultsforcobipartite graphs,bipartitegraphsandtrees.InSections4and5,weproveourresultsforcographsandsplitgraphs,respectively.In Section 5we alsoshow that our NP-hardness reductionfor splitgraphs canbe usedto provethat the threecontraction blockersproblems,restrictedtosplitgraphs,are W[1]-hardwhenparameterizedby d.Thelatterresultmeansthatforsplit graphs theseproblems areunlikely to be fixed-parameter tractable withparameter d. In Sections6 and7 we prove our resultsforintervalgraphsandchordalgraphs,respectively.
Sections8and9containtheresultsmentionedinPart II.InSection8wefirstprovedichotomiesforthethreecontraction blockerandthreevertexblockerproblemswhenweclassify onbasis ofthesizeof
π
∈ {
α
,
χ
,
ω
}
.Inthesamesection,we modifythehardness constructionfor 1-ContractionBlocker(ω
)toprovethat 1-ContractionBlocker(ω
) is NP-complete evenforC4-freeperfectgraphswithω
=
3.InSection9weproveTheorem2.Section10containsanumberofopenproblemsanddirectionsforfutureresearch.
2. Preliminaries
We only consider finite, undirected graphs that have no self-loops and no multiple edges; we recall that when we contractanedge,noself-loopsormultipleedgesarecreated.Wereferto [14] or [45] forundefinedterminologyandto [15] formoreonparameterizedcomplexity.
LetG
= (
V,
E)
be agraph.Forasubset S⊆
V ,we let G[
S]
denotethe subgraphof G induced by S,whichhasvertex set S andedgeset{
uv∈
E|
u,
v∈
S}
.Wewrite H⊆
iG ifa graphH isan inducedsubgraphof G. Moreover,foravertex v∈
V ,wewriteG−
v=
G[
V\ {
v}]
andforasubsetV⊆
V wewrite G−
V=
G[
V\
V]
.Foraset{
H1,
. . . ,
Hp}
ofgraphs, G is(
H1,
. . . ,
Hp)
-free ifG hasnoinducedsubgraphisomorphictoagraphin{
H1,
. . . ,
Hp}
;ifp=
1 wemaywrite H1-freeinsteadof
(
H1)
-free.Thecomplement ofG isthegraphG= (
V,
E)
withvertexset V andanedgebetweentwoverticesuandv ifandonlyif uv
∈
/
E.Recallthatthecontractionofanedgeuv∈
E removestheverticesu and v fromthegraph G andreplaces themby a newvertexthat ismade adjacenttoprecisely those verticesthat were adjacentto u or v in G.Thisnewgraphwillbedenotedby G
|
uv.Inthatcasewemayalsosaythatu iscontractedonto v,andweusev todenote thenewvertexresultingfromthe edgecontraction.Thesubdivision of anedge uv∈
E removestheedgeuv from G andreplacesitbyanewvertexw andtwoedgesu w andw v.
Let G and H betwo vertex-disjointgraphs. The join operation
⊗
addsan edge betweenevery vertexof G andevery vertexof H .The union operation⊕
takesthe disjointunion of G and H ,that is, G⊕
H= (
V(
G)
∪
V(
H),
E(
G)
∪
E(
H))
. We denote the disjointunion of p copies of G by pG. For n≥
1, the graph Pn denotes the path on n vertices,that is, V(
Pn)
= {
u1,
. . . ,
un}
and E(
Pn)
= {
uiui+1|
1≤
i≤
n−
1}
.Forn≥
3,thegraphCn denotes thecycle onn vertices,that is, V(
Cn)
= {
u1,
. . . ,
un}
and E(
Cn)
= {
uiui+1|
1≤
i≤
n−
1}
∪ {
unu1}
.ThegraphC3isalsocalledthetriangle.Theclaw K1,3isthe4-vertexstar,thatis,thegraphwithverticesu, v1,v2, v3andedgesuv1,uv2,uv3.
LetG
= (
V,
E)
bea graph.Asubset K⊆
V iscalledaclique of G ifanytwo verticesin K areadjacenttoeach other. Thecliquenumberω
(
G)
isthe numberofverticesinamaximumclique ofG.Asubset I⊆
V iscalledanindependentsetof G ifanytwoverticesin I are non-adjacent toeach other.Theindependencenumber
α
(
G)
isthenumberofverticesin amaximumindependentsetofG.Forapositiveintegerk, ak-colouring of G isamappingc:
V→ {
1,
2,
. . . ,
k}
suchthatc
(
u)
=
c(
v)
wheneveruv∈
E.Thechromaticnumberχ
(
G)
isthesmallestintegerk forwhichG hasak-colouring.Asubset ofedges M⊆
E iscalledamatching ifno twoedgesof M shareacommonend-vertex.Thematchingnumberμ
(
G)
isthe numberofedgesinamaximummatchingofG.Avertexv suchthat M containsanedgeincidentto v issaturated byM;otherwise v isunsaturated byM.Asubset S
⊆
V isavertexcover ofG ifeveryedgeofG isincidenttoatleastonevertex of S.The Coloring problemisthatoftestingifagraphhasak-colouringforsomegiveninteger k.Theproblems Clique and IndependentSetarethoseoftestingifagraphhasacliqueorindependentset,respectively,ofsizeatleast k.The Vertex Cover problemisthat oftestingifagraphhasavertexcoverofsize atmost k. Weneedthefollowinglemmaatseveral placesinourpaper.
Lemma1([41]).VertexCoveris NP-completeforC3-freegraphs.
Welet
(
∅,
∅)
betheempty graph.Anintervalgraph isagraphsuchthatonecanassociateanintervalofthereallinewith every vertexsuchthat twoverticesareadjacentifandonlyifthecorrespondingintervalsintersect.Agraphiscobipartiteifitisthecomplementofabipartite (2-colourable)graph.Agraphischordal ifitcontainsnoinducedcycleonmorethan threevertices.Agraphisasplitgraph ifithasasplitpartition,whichisapartitionofitsvertexsetintoaclique K andan independentset I.Splitgraphscoincidewith
(
2P2,
C4,
C5)
-freegraphs [18].A P4-freegraphisalsocalledacograph.Agraphisperfect ifthechromaticnumberofeveryinducedsubgraphequalsthesizeofalargestcliqueinthatsubgraph. Ahole is aninduced cycleonatleastfiveverticesandanantihole isthecomplementofahole.Aholeorantiholeisodd
ifitcontainsan oddnumberofvertices.Weneedthefollowingwell-knowntheoremofChudnovsky,Robertson,Seymour, andThomas.ThistheoremcanalsobeusedtoverifythattheothergraphclassesinTable1areindeedsubclassesofperfect graphs.
Theorem3(StrongPerfectGraphTheorem[9]).Agraphisperfectifandonlyifitcontainsnooddholeandnooddantiholeasan inducedsubgraph.
3. Cobipartitegraphs,bipartitegraphsandtrees
Wefirstconsiderthecontractionblockerproblemsandthenthedeletionblockerproblems.
3.1. Contractionblockers
Ourfirstresultisahardnessresultforcobipartitegraphsthatfollowsdirectlyfromaknownresult.
Theorem4.1-ContractionBlocker(
α
)is NP-completeforcobipartitegraphs.Proof. Golovach,Heggernes,van ’t HofandPaul [21] consideredthe s-ClubContractionproblem.Recallthatthisproblem takesasinputa graph G andan integerk andaskswhetherG canbek-contracted intoagraphwithdiameteratmost s for some fixed integer s. They showed that 1-ClubContraction is NP-complete even forcobipartite graphs. Graphs of diameter 1 are complete graphs, that is, graphs with independence number 1, whereas cobipartite graphs that are not completehaveindependencenumber 2.
2
We nowfocuson
π
=
χ
andπ
=
ω
.Forournextresult(Theorem5) weneedsomeadditionalterminology.Abicliqueisacompletebipartitegraph,whichisnontrivial ifithasatleastoneedge.Abicliquevertex-partition ofagraphG
= (
V,
E)
isa set
S
ofmutuallyvertex-disjoint bicliquesin G suchthatevery vertexofG iscontainedinoneofthebicliquesofS
. The BicliqueVertex-PartitionproblemconsistsintestingwhetheragivengraphG hasabicliquevertex-partitionofsizeat most k,forsomepositiveintegerk.Fleischneretal. [17] showedthatthisproblemis NP-completeevenforbipartitegraphs andk=
3.WearenowreadytoproveTheorem5.
Theorem5.For
π
∈ {
χ
,
ω
}
, ContractionBlocker(π
)is NP-completeforcobipartitegraphs.Proof. Sincecobipartitegraphsareperfectandclosedunderedgecontractions,wemayassumewithoutlossofgenerality that
π
=
χ
.The problemisin NP, as Coloring ispolynomial-timesolvableon cobipartitegraphs andthen we cantake the sequence ofedge contractions ascertificate. We reduce from BicliqueVertex-Partition. Recallthat thisproblem is NP-completeevenforbipartitegraphsandk=
3 [17].Astheproblemispolynomial-timesolvableforbipartitegraphsandk
=
2 (see [17]),wemayaskforabicliquevertex-partitionofsizeexactly 3,inwhicheachbicliqueisnontrivial.Let(G
,
3) be an instanceof BicliqueVertex-Partition, whereG is a connectedbipartite graphonn verticesthat has partition classes X and Y .We claimthat G hasa bicliquevertex-partitionconsistingofthree non-trivialbicliquesifand onlyifG canbe(
n−
6)
-contractedintoagraphGwithχ
(
G)
≤
3 (sod=
χ
(
G)
−
3).FirstsupposethatG hasabicliquevertex-partition
S
ofsize3.LetS1,
S2,
S3bethethree(nontrivial)bicliquesinS
.Let Ai,
Bi bethe twobipartitionclassesof Si fori=
1,
2,
3.So,in G,we havethat A1,
A2,
A3, B1,
B2,
B3 are sixcliques thatpartitionthe verticesof G,andmoreover,thereisnoedgebetweenavertexof Ai anda vertexof Bi,fori
=
1,
2,
3.In Gwecontract eachclique Ai toasingle vertexthatwe givecolour i,andwecontracteachclique Bi toasingle vertexthat
wegivecolour i aswell.Inthiswaywehaveobtaineda6-vertexgraphG(sothenumberofcontractionsisn
−
6)witha 3-colouring.Thus,χ
(
G)
≤
3.Now suppose that G can be
(
n−
6)
-contracted into a graph G withχ
(
G)
≤
3. We first observe that the class of cobipartite graphsis closedunder takingedge contractions; indeed,ife is an edge connectingtwo verticesof thesame partition class,thencontractinge results inasmallerclique,andife isanedge connectingtwoverticesoftwo different partition classes, then contracting e is equivalent to removing one of its end-vertices and making its other end-vertex adjacenttoeveryothervertexintheresultinggraph.Astheclassofcobipartitegraphsisclosedundertakingcontractions,Giscobipartite.Ascobipartitegraphshave inde-pendencenumberatmost 2,eachcolourclassinacolouringofG musthavesizeatmost 2.Consequently,G musthave exactlysixverticesa1
,
a2,
a3,b1,
b2,
b3 suchthat a1,
a2,
a3 formaclique,b1,
b2,
b3 formaclique,andmoreover,ai andbiare notadjacent,for i
=
1,
2,
3.Thismeans that wedid notcontract an edge uv with u∈
X and v∈
Y (as theresulting vertexwouldbeadjacenttoallothervertices).Hence,wemayassumewithoutlossofgeneralitythatfori=
1,
2,
3,each aicorresponds to a setofvertices Ai
⊂
X (that we contractedinto thesingle vertexai) andthat each bi corresponds toasetofvertices Bi
⊂
Y (thatwecontractedintothesingle vertexbi).Aseach pairai,
bi isnon-adjacent,novertexof Ai isadjacentto avertexof Bi.Consequently,in G,we findthat eachset Ai
∪
Bi induces abiclique. Hence, thesets A1∪
B1, A2∪
B2and A3∪
B3 formabicliquevertex-partitionofG thathassize 3.2
Wenowassumethatd isfixed.Weshowthat d-ContractionBlocker(
π
)becomespolynomial-timesolvableon cobipar-titegraphsforπ
∈ {
χ
,
ω
}.
Forπ
=
χ
,wecanprovethisevenfortheclassofgraphswithindependencenumberatmost 2, orequivalently,theclassof3P1-freegraphs,whichproperlycontainstheclassofcobipartitegraphs.Theorem6.Foranyfixedd
≥
0,the d-ContractionBlocker(χ
)problemcanbesolvedinpolynomialtimefor3P1-freegraphs.Proof. LetG beagraphwith
α
(
G)
≤
2.Consideracolouringwithχ
(
G)
colours.Thesizeofeverycolourclassisatmost2. HenceeverysubgraphofG inducedbytwocolourclasseshasatmost4 vertices,andassuchhasaspanningforestwithin totalatmost3 edges.Thismeansthat wecancontracttwocolourclassestoanindependentset(thatis,toa newcolour class)byusingatmostthreeedgecontractions.Thisobservationgivesusthefollowingalgorithm.Weconsidereach setof atmostthreeedgecontractions,whichweperform.Afterwardswedecreased by 1andrepeatthisprocedureuntild=
0. ForeachresultinggraphG obtainedinthiswaywecheck whetherχ
(
G)
≤
χ
(
G)
−
d.Ifso,thenthealgorithmreturnsa yes-answer.Otherwise,thatis,ifχ
(
G)
>
χ
(
G)
−
d foreachresultinggraph G,ouralgorithmreturnsano-answer.Letm bethenumberofedges ofG.Thenthetotalnumberofresultinggraphs Gisatmostm3d,whichispolynomial asd isfixed.Because Coloring ispolynomial-timesolvableongraphswithindependencenumberatmost 2 [28] andthis classisclosedunderedgecontractions,ouralgorithmrunsinpolynomialtime.
2
Corollary1.Let
π
∈ {
χ
,
ω
}
.Foranyfixedd≥
0,the d-ContractionBlocker(π
)problemcanbesolvedinpolynomialtimeoncobipartitegraphs.
Proof. For
π
=
χ
thisfollowsimmediatelyfromTheorem6.Ascobipartitegraphsareperfectandclosedunderedge con-traction,weobtainthesameresultforπ
=
ω
.2
Wenowconsidertheclassofbipartitegraphs.If
π
∈ {
χ
,
ω
}
,then ContractionBlocker(π
)istrivialforbipartitegraphs (andthus alsofortrees).Tothe contrary,forπ
=
α
,we willshow that ContractionBlocker(π
) is NP-hardforbipartite graphs. The complexity of d-ContractionBlocker(α
) remains open forbipartite graphs. Bipartite graphs are not closed underedgecontraction.Therefore membershipto NP cannotbeestablished bytakinga sequenceofedgecontractions as thecertificate,eventhough dueto König’sTheorem(see,forexample,[14]), IndependentSetispolynomial-timesolvable forbipartitegraphs.Theorem7.ContractionBlocker(
α
)is NP-hardonbipartitegraphs.Proof. WeknowfromTheorem4that 1-ContractionBlocker(
α
)is NP-completeon cobipartitegraphs.Consider a cobi-partitegraphG withm edgesandan integerk,whichtogetherforman instanceof 1-ContractionBlocker(α
).Subdivide eachofthem edgesofG inorderto obtainabipartitegraphG.We claimthat(
G,
k)
isayes-instanceof 1-Contraction Blocker(α
)ifandonlyif(
G,
α
(
G)
−
1,
k+
m)
isayes-instanceof ContractionBlocker(α
).Firstsupposethat
(
G,
k)
isa yes-instanceof 1-ContractionBlocker(α
).InG wefirstperformm edgecontractionsto get G back.Wethenperformk edgecontractionstogetindependencenumber1=
α
(
G)
−(
α
(
G)
−
1)
.Hence,(
G,
α
(
G)
−
1,
Now suppose that
(
G,
α
(
G)
−
1,
k+
m)
isayes-instanceof ContractionBlocker(α
).Then thereexistsa sequenceofk
+
m edge contractions that transform G intoa complete graph K . We may assume that K hassize at least 4(aswe could haveaddedwithoutlossofgeneralitythreedominatingverticestoG withoutincreasingk).As K hassizeatleast 4, each subdividededgemustbecontractedbacktotheoriginaledgeagain.Thisoperationcostsm edgecontractions,sowe contract G toK usingatmostk edgeoperations.Hence,(
G,
k)
isayes-instanceof 1-ContractionBlocker(α
).Thisproves theclaimandhencethetheorem.2
WecomplementTheorem7byshowingthat ContractionBlocker(
α
)islinear-timesolvableontrees.Inordertoprove thisresultwemakeaconnectiontothematchingnumberμ
ofagraph.Theorem8.ContractionBlocker(
α
)islinear-timesolvableontrees.Proof. Let
(
T,
d,
k)
beaninstanceof ContractionBlocker(α
),whereT isatreeonn vertices.Wefirstdescribeour algo-rithm andproveits correctness.Afterwards,we analyzeits runningtime.Throughouttheprooflet M denoteamaximum matchingofT .As
α
(
T)
+
μ
(
T)
=
n byKönig’sTheorem(see,forexample,[14]),we findthat(
T,
d,
k)
isano-instanceifd>
n−
μ
(
T)
. Assumethatd≤
n−
μ
(
T)
.Weobservethattreesareclosedunderedgecontraction.Hence,contractinganedgeofT resultsinanewtreeT.Moreover, Thasn
−
1 verticesandtheedgecontractionneitherincreasedtheindependencenumbernor thematchingnumber.Asα
(
T)
+
μ
(
T)
=
n andsimilarlyα
(
T)
+
μ
(
T)
=
n−
1,thismeansthateitherα
(
T)
=
α
(
T)
−
1 orμ
(
T)
=
μ
(
T)
−
1.Firstsupposethatd
≤
n−
2μ
(
T)
.Thereareexactlyσ
(
T)
=
n−
2μ
(
T)
verticesthatareunsaturatedbyM. Letuv bean edge,suchthatu isunsaturated.AsM ismaximum,v mustbesaturated.Then,bycontractinguv,weobtainatreeTsuch thatμ
(
T)
=
μ
(
T)
.Itfollowsfromtheabovethatα
(
T)
=
α
(
T)
−
1.Saythatwecontractedu ontov.TheninT wehave that v issaturatedby M,whichisamaximummatchingofT aswell.Thus,ifd≤
n−
2μ
(
T)
,contractingd edges,oneof the end-vertices ofwhichis unsaturatedby M, yields atree T withμ
(
T)
=
μ
(
T)
andα
(
T)
=
α
(
T)
−
d.Since an edge contractionreducestheindependencenumberbyatmost 1,itfollowsthatthisisoptimal.Hence,asd≤
n−
2μ
(
T)
,wefind that(
G,
T,
k)
isayes-instanceifk≥
d andano-instanceifk<
d.Now supposethatd
>
n−
2μ
(
T)
.Supposethatwefirstcontractthen−
2μ
(
T)
edgesthathaveexactlyoneend-vertex that isunsaturatedby M.Itfollowsfromtheabove thatthisyields atreeT withμ
(
T)
=
μ
(
T)
andα
(
T)
=
α
(
T)
− (
n−
2
μ
(
T))
.SinceT doesnotcontainanyunsaturatedvertex, M isaperfectmatchingofT.Then,contractinganyedgeinTresultsinatree T with
μ
(
T)
=
μ
(
T)
−
1 andthus,α
(
T)
=
α
(
T)
.Ifwe contractanedge uv∈
M,theresultingvertexuv isunsaturatedby M
=
M\ {
uv}
in T.Hence, asexplainedabove,ifinadditionwe contractnowanedge(
uv)
w,we obtainatreeT withα
(
T)
=
α
(
T)
−
1 andμ
(
T)
=
μ
(
T)
.Repeatingthisprocedure,wemayreducetheindependence number of T byd with n−
2μ
(
T)
+
2(
d−
n+
2μ
(
T))
=
2(
d+
μ
(
T))
−
n edge contractions.Below we show that thisis optimal.Supposethatwecontractp edgesinT .LetTbetheresultingtree.Wehave
α
(
T)
+
μ
(
T)
=
n−
p.Asμ
(
T)
≤
12(
n−
p)
, thismeansthatα
(
T)
≥
12(
n−
p)
.Ifp<
2(
d+
μ
(
T))
−
n wehave−
p2>
−(
d+
μ
(
T))
+
n2,andthusα
(
T)
≥
12(
n−
p)
>
n2−
d−
μ
(
T)
+
n2=
α
(
T)
−
d.
Soatleast2
(
d+
μ
(
T))
−
n edgecontractionsarenecessarytodecreasetheindependencenumberbyd.Itremainstocheck ifk issufficientlyhighforustoallowthisnumberofedgecontractions.As we can find a maximum matching of tree T (and thus compute
μ
(
T)
) in O(
n)
time by using the algorithm of Savage [43],ouralgorithmrunsinO(
n)
time.2
Remark1.ByKönig’sTheorem,wehavethat
α
(
G)
+
μ
(
G)
= |
V(
G)
|
foranybipartitegraphG,butwecanonlyusetheproof ofTheorem8to obtainaresultfortrees forthefollowingreason:treesformthelargestsubclassof(connected)bipartite graphsthatareclosedunderedgecontraction,andthispropertyplaysacrucialroleinourproof.3.2. Deletionblockers
We firstshow thatall threedeletion blockerproblemsarepolynomial-timesolvableforbipartite graphs(andthus for trees).Itisknownalreadythat DeletionBlocker(
α
)ispolynomial-timesolvableforbipartitegraphs [3,13].Henceitsuffices to provethatthesameholdsfor DeletionBlocker(π
) whenπ
∈ {
χ
,
ω
}
.Inordertodosoweneedthefollowingrelation between 1-DeletionBlocker(ω
)and VertexCover.Proposition1.LetG beagraphwithatleastoneedgeandletk
≥
1 beaninteger.Then(
G,
ω
(
G)
−
1,
k)
isayes-instanceof DeletionProof. Let G
= (
V,
E)
be a graph with|
E|
≥
1. Thus,ω
(
G)
≥
2. Letk≥
1 be an integer. First suppose that(
G,
k)
is a yes-instanceof VertexCover,thatis,G hasavertexcoverVofsizeatmostk. So,every edgeofG isincidenttoatleast one vertexof V.Then, deletingall verticesof V yieldsa graph G withno edges.This meansthatω
(
G)
≤
1,andthus(
G,
ω
(
G)
−
1,
k)
isayes-instancefor DeletionBlocker(ω
).Nowsupposethat(
G,
ω
(
G)
−
1,
k)
isayes-instanceof Deletion Blocker(ω
).ThenthereexistsasetV⊆
V ofsize|
V|
≤
k suchthatω
(
G−
V)
≤
1.ThisimpliesthatG−
Vhasnoedges. Thus VisavertexcoverofG ofsizeatmostk.So,(
G,
k)
isayes-instancefor VertexCover.2
Proposition1hasthefollowingcorollary,whichwewillapplyinthissectionandatsomeotherplacesinourpaper.
Corollary2.LetG be atriangle-freegraph withatleastoneedgeandletk
≥
1 beaninteger.Then(
G,
k)
isayes-instance of1-DeletionBlocker(
ω
)ifandonlyif(
G,
k)
isayes-instanceof VertexCover. Wearenowreadytoprovethefollowingresult.Theorem9.For
π
∈ {
χ
,
ω
}
, DeletionBlocker(π
)canbesolvedinpolynomialtimeonbipartitegraphs.Proof. Asbipartite graphsareperfectandclosedundervertexdeletion,theproblems DeletionBlocker(
ω
) and Deletion Blocker(χ
) are equivalent. Therefore, we only have to consider the case whereπ
=
ω
. As bipartite graphs have clique numberatmost 2, DeletionBlocker(ω
) and 1-DeletionBlocker(ω
) are equivalent. Asbipartite graphs are triangle-free, we can apply Corollary 2. To solve VertexCoveron bipartite graphs, König’sTheorem tells usthat it suffices to find a maximummatching,whichtakesO(
n2.5)
timeonn-vertexbipartitegraphs [27].2
Wenowconsiderthe classofcobipartitegraphs.Itisknownthat DeletionBlocker(
π
)ispolynomial-timesolvableon cobipartitegraphsifπ
∈ {
ω
,
χ
}
[13].Henceweonlyhavetodealwiththecaseπ
=
α
.Forthiscaseweprovethefollowing result,whichfollowsimmediatelyfromTheorem9byconsideringthecomplement.Theorem10. DeletionBlocker(
α
)canbesolvedinpolynomialtimeoncobipartitegraphs.4. Cographs
It is well known (see forexample [8]) that a graph G isa cograph ifandonly if G can be generatedfrom K1 by a
sequenceofoperations, whereeachoperation iseithera joinoraunion operation.RecallfromSection 2that we denote theseoperationsby
⊗
and⊕
,respectively.SuchasequencecorrespondstoadecompositiontreeT ,whichhasthefollowing properties:1. itsrootr correspondstothegraphGr
=
G;2. every leaf x of T corresponds to exactly one vertex of G, and vice versa, implying that x corresponds to a unique single-vertexgraphGx;
3. every internal node x of T hasatleast twochildren, iseither labelled
⊕
or⊗
, andcorresponds toan induced sub-graph GxofG definedasfollows:•
ifx isa⊕
-node,thenGx isthedisjointunionofallgraphsGy where y isachildofx;•
ifx isa⊗
-node,thenGx isthejoinofallgraphsGy where y isachildofx.A cograph G may have morethan one such tree buthas exactlyone uniquetree [11], called thecotree TG of G,ifthe
followingadditionalpropertyisrequired:
4. Labelsofinternalnodesonthe(unique)pathfromanyleaftor alternatebetween
⊕
and⊗
.Notethat TG has O
(
n)
vertices.Forourpurposeswe mustmodify TG byapplyingthefollowingknownprocedure(seeforexample [6]).Wheneveran internal node x of TG hasmorethan two children y1 and y2,we remove the edges xy1
and xy2andaddanewvertexxwithedges xx,xy1 andxy2.If x isa
⊕
-node,thenxisa⊕
-node,andifx isa⊗
-node,thenxisa
⊗
-node.Applyingthisruleexhaustivelyyieldsatreeinwhicheachinternalnodehasexactlytwochildren.We denotethistreebyTG.BecauseTG has O(
n)
vertices,modifying TG intoTG takeslineartime.Corneil, PerlandStewart [12] provedthat theproblemofdecidingwhethera graphwithn verticesandm edges isa cographcanbesolvedintime O
(
n+
m)
.Theyalsoshowedthatinthesametimeitispossibletoconstructitscotree(ifit exists).AsmodifyingTG intoTG takes O(
n+
m)
time,weobtainthefollowinglemma.Lemma2.LetG beagraphwithn verticesandm edges.DecidingifG isacographandconstructingTG (ifitexists)canbedonein timeO
(
n+
m)
.Fortwointegersk and
wesaythatagraphG canbe
(
k,
)
-contracted intoagraphH ifG canbemodifiedintoH bya sequencecontainingk edgecontractionsandvertexdeletions.Notethatcographsareclosedunderedgecontractionand undervertexdeletion.Infact,toproveourresultsforcographs,wewillprovethefollowingmoregeneralresult.
Theorem11.Let
π
∈ {
α
,
χ
,
ω
}
.Theproblemofdeterminingthelargestinteger d suchthatacographG withn verticesandm edges canbe(
k,
)
-contractedintoacographH withπ
(
H)
≤
π
(
G)
−
d canbesolvedinO(
n2+
mn+ (
k+ )
3n)
time.Proof. Firstconsider
π
=
α
.LetG beacographwithn verticesandm edgesandletk,
betwopositiveintegers.Wefirst constructTG.WethenconsidereachnodeofTG byfollowingabottom-upapproachstartingattheleavesofTG andending initsrootr.
Let x be a node of TG. Recall that Gx is the subgraph ofG induced by all vertices that corresponds to leavesin the
subtree of TG rooted at x. With node x we associatea table that records the following data: for each pair of integers
i
,
j≥
0 with i+
j≤
k+
we compute thelargest integer d suchthat Gx can be(
i,
j)
-contracted intoa graph Hx withα
(
Hx)
≤
α
(
Gx)
−
d.Wedenotethisintegerd byd(
i,
j,
x)
.Leti,
j≥
0 withi+
j≤
k+
. Case1. x isaleaf.Then Gx isa1-vertexgraphmeaningthatd
(
i,
j,
x)
=
0 if j=
0,whereasd(
i,
j,
x)
=
1 if j≥
1. Case2. x isa⊕
-node.Let y andz bethetwochildrenofx.Then,asGx isthedisjointunionofGy andGz,wefindthat
α
(
Gx)
=
α
(
Gy)
+
α
(
Gz)
.Hence,wehave
d
(
i,
j,
x)
=
max{
α
(
Gx)
− (
α
(
Gy)
−
d(
a,
b,
y)
+
α
(
Gz)
−
d(
i−
a,
j−
b,
z))
|
0≤
a≤
i,
0≤
b≤
j}
=
max{
d(
a,
b,
y)
+
d(
i−
a,
j−
b,
z)
|
0≤
a≤
i,
0≤
b≤
j}.
Case3. x isa
⊗
-node.Since x isa
⊗
-node, Gx isconnectedandassuchhasaspanningtreeT .Ifi+
j≥ |
V(
Gx)
|
and j≥
1,thenwecancontract i edgesof T inthegraph Gx followedby j vertexdeletions.As eachoperationwillreduce Gx byexactlyone vertex,thisresultsintheemptygraph.Hence,d
(
i,
j,
x)
=
α
(
Gx)
.Fromnowonassumethati+
j<
|
V(
Gx)
|
or j=
0.Assuch,anygraphwe can obtain from Gx by using i edge contractions and j vertexdeletions is non-empty and hencehas independence
numberatleast 1.
Let y andz bethetwochildrenofx.Then,asGx isthejoinofGyandGz,wefindthat
α
(
Gx)
=
max{
α
(
Gy),
α
(
Gz)
}
.Inordertodetermined
(
i,
j,
x)
wemustdosomefurtheranalysis.LetS beasequencethatconsistsofi edgecontractionsandj vertexdeletionsofGx such thatapplying S onGx resultsina graphHx with
α
(
Hx)
=
α
(
Gx)
−
d(
i,
j,
x)
.We partition Sinto five sets Sey, Sez, Seyz, Svy, Szv, respectively, asfollows. Let Sey and Sez be the set ofcontractions ofedges withboth end-vertices in Gy andwithboth end-vertices in Gz, respectively. Let Seyz be the setof contractions of edges withone
end-vertex in Gy andtheother one in Gz (notice that, without lossof generality, thenew vertexresultingfrom such a
contraction maybe consideredto belong to Gy).Letay
= |
Sey|
andletaz= |
Sze|
.Then|
Seyz|
=
i−
ay−
az.Let Svy and Szvbe thesetofdeletionsofverticesinGy andGz,respectively.Letb
= |
Svy|
.Then|
Svz|
=
j−
b.Wedistinguishbetweentwocases.
Firstassumethat Se
yz
= ∅
.Thenay+
az=
i.LetHybethegraphobtainedfromGy afterapplyingthesubsequenceofS,consistingofoperationsinSe
y
∪
Svy,onGy.LetHzbedefinedanalogously.Thenwehaveα
(
Hx)
=
max{
α
(
Hy),
α
(
Hz)
}
=
max{
α
(
Gy)
−
d(
ay,
b,
y),
α
(
Gz)
−
d(
az,
j−
b,
z)
}
=
max{
α
(
Gy)
−
d(
ay,
b,
y),
α
(
Gz)
−
d(
i−
ay,
j−
b,
z)
},
wherethesecondequalityfollowsfromthedefinitionofS.
Nowassumethat Seyz
= ∅
.Recallthati+
j<
|
V(
Gx)
|
or j=
0.Henceα
(
Hx)
≥
1.Ourapproachisbasedonthefollowingobservations.
First, contractingan edge with one end-vertex in Gy andthe other one in Gz is equivalent to removing these two
end-verticesandintroducinganewvertexthatisadjacenttoallotherverticesofGx (suchavertexissaidtobeuniversal).
Second,assumethat Gy containstwodistinctverticesu anduandthatGz containstwodistinctverticesv and v.Now
supposethatwearetocontracttwoedgesfrom
{
uv,
uv,
uv,
uv}
.Contractingtwoedgesofthissetthathaveacommon end-vertex, sayedgesuv and uv,isequivalenttodeleting u,
v,
v fromGx andintroducing anewuniversalvertex.Con-tracting two edges with no common end-vertex, say uv and uv, is equivalent to deleting all four vertices u
,
u,
v,
vfrom Gx and introducing two new universal vertices. Because the two new universal vertices in the latter choice are
adjacent,whereasthevertexumaynotbeuniversalaftermakingtheformerchoice,thelatterchoicedecreasesthe inde-pendence numberbythesameoralargervaluethantheformerchoice.Hence,wemayassume withoutlossofgenerality that thelatterchoicehappened.Moregenerally,thecontractededgeswithoneend-vertexinGy andtheotheronein Gz
can beassumedtoformamatching.Wealsonote thatintroducing anewuniversalvertextoagraphdoesnotintroduce anynewindependentsetotherthanthesingletonsetcontainingthevertexitself.