Contraction and deletion blockers for perfect graphs and H-free graphs

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

,

∗

a_{Kadir Has University, Istanbul, Turkey} b_{Durham University, Durham, UK} c_{CNAM, Laboratoire CEDRIC, Paris, France} d_{University 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 via

at most

k graph

operations from

some fixed set S?

As parameters we take the chromatic number

χ

, clique number

ω

and independence number

α

, 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} and

π

∈ {

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} and

π

∈ {

χ

,

ω

,

α

}

restricted to H-free

graphs.

1. Introduction

AtypicalgraphmodificationproblemaimstomodifyagraphG,viaasmallnumberofoperationsfromaspecifiedset S, intosome othergraph H thathasacertain desiredproperty,whichusually describesa certaingraphclass

G

towhich H

mustbelong.Inthiswayavarietyofclassicalgraph-theoreticproblemsiscaptured.Forinstance,ifonlyk vertexdeletions areallowedandH mustbeanindependentsetoraclique,weobtainthe IndependentSetor Clique problem,respectively. Now,insteadoffixing aparticular graphclass

G

,wefixacertaingraphparameter

π

.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 atmost

k 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:[email protected](Ö.Y. Diner),[email protected](D. Paulusma),[email protected](C. Picouleau),

[email protected](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

,thecliquenumber

ω

andthechromaticnumber

χ

.

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,inthispaper

π

belongsto

{

α

,

ω

,

χ

}

.

Wearenowreadytodefineourdecisionproblemsinageneralway: ContractionBlocker(

π

)

Instance: a graph G and two integers d

,

k

≥

0

Question: 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

≥

0

Question: 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

) and

d-DeletionBlocker(

π

)_{,respectively.}

d-ContractionBlocker(

π

)

Instance: a graph G and an integer k

≥

0

Question: 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

≥

0

Question: 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

)_for

π

∈ {

ω

_,

χ

_,

α

}.

_{Inorder todoso, wewill alsoconsiderthe problems d-Contraction}Blocker(

π

)_and

d-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.Thelatterauthorsalsoprovedthatfor

π

∈ {

ω

,

χ

}

, DeletionBlocker(

π

) _{is polynomial-time solvable for cobipartite graphs. Moreover, they showed that for}

π

∈ {

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 K_r 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 HadwigerNumber_{inanotherway:the Club}Contraction_{problem(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

, the

H

-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 in

H

. For instance, the problem

{

K2

}

-transversal_{isthesameas Vertex}Cover_{.Herearesomeexamplesofspecificconnectionsbetweengraphtransversals} andblockerproblems.

– Let

H

be thefamily

{

Kp

|

p

≥

2

}

ofall completegraphsonatleasttwovertices.Then

H

-transversal_{isequivalentto} 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 OddCycleTransversal_{problemis 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 eachmaximumsetsatisfyingsomespecifiedproperty

π

byatleastd vertices.Forinstance,ifthepropertyisbeingan independentset, 1-Transversal isequivalentto 1-DeletionBlocker(

α

).

3. BipartiteContraction. Theproblem BipartiteContraction_{istotestwhetheragraphcan 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 hasalsobeenstudiedwhen

π

=

χ

.Avertexofagraph G iscritical

ifits 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 Deletion}Blocker(

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-completefor

π

∈ {

α

,

ω

,

χ

}

even when re-strictedtospecialgraphclasses.Non-surprisingly, ContractionBlocker(

π

)is NP-complete for

π

∈ {

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 Deletion}Blocker(

π

)_{belong to NP whenrestrictedtoagraphclass}

G

_{.However,when}

G

_is closedunderedgecontractionorvertexdeletion,respectively,and

π

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

and

π

=

χ

(notethat

ω

=

χ

holdsbydefinitionofaperfectgraph).Another

Table 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 thisfor

χ

andthusfor

ω

,whereas theresultfor

α

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

and

π

=

χ

even forthe perfectgraphclasses inthis table that are not closed underedgecontraction,namelytheclassesofbipartitegraphs;C4-freeperfectgraphswithcliquenumber 3;andtheclass

ofperfectgraphsitself.Astheclassofperfectgraphsisnotclosedunderedgecontractionwecouldforperfectgraphsonly deducethatthethreecontractionblockerproblemsare NP-hard(evenifd

=

1).Astheclassofcographscoincideswiththe classofP4-freegraphs(where Prdenotesther-vertexpath)andsplitgraphsare P5-free,thecorrespondingrowsinTable1

show a complexity jumpofall ourproblemsfor Pt-free graphsfromt

=

4 tot

=

5. Recallalso fromSection 1.1that the

HadwigerNumber_{problemisaspecialcaseof Contraction}Blocker(

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

)dependingonthesizeof

π

,thatis,weprovethefollowing theorem.

Theorem1.Thefollowingsixdichotomieshold:

(i) ContractionBlocker(

α

)_{ispolynomial-timesolvableforgraphswith}

α

=

_{1 and 1-Contraction}Blocker(

α

)_{is NP-complete}

forgraphswith

α

=

2;

(ii) ContractionBlocker(

χ

)ispolynomial-timesolvableforgraphswith

χ

=

2 and 1-ContractionBlocker(

χ

)is NP-complete

forgraphswith

χ

=

3;

(iii) ContractionBlocker(

ω

)_{ispolynomial-timesolvableforgraphswith}

ω

=

_{2 and 1-Contraction}Blocker(

ω

)_{is NP-complete}

forgraphswith

ω

=

3;

(iv) DeletionBlocker(

α

)ispolynomial-timesolvableforgraphswith

α

=

1 and 1-DeletionBlocker(

α

)is NP-completeforgraphs

with

α

=

2;

(v) DeletionBlocker(

χ

)_{ispolynomial-timesolvableforgraphswith}

χ

=

_{2 and 1-Deletion}Blocker(

χ

)_{is NP-completeforgraphs}

with

χ

=

3;

(vi) DeletionBlocker(

ω

)ispolynomial-timesolvableforgraphswith

ω

=

1 and 1-DeletionBlocker(

ω

)is NP-completeforgraphs

with

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 with

ω

=

3 and other results ofTable 1 to

prove 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

⊕

denotesthedisjointunionof

Theorem2.LetH beagraph.Thenthefollowingholds:

(i) IfH

⊆

iP4,then DeletionBlocker(

α

)ispolynomial-timesolvableforH -freegraphs,otherwiseitis NP-hardorco-NP-hardfor

H -freegraphs.

(ii) If H

⊆

i P4,then ContractionBlocker(

α

)ispolynomial-timesolvableforH -freegraphs,otherwiseitis NP-hardforH -free

graphs.

(iii) IfH

⊆

iP4,then DeletionBlocker(

ω

)_{ispolynomial-timesolvableforH -freegraphs;otherwiseitis NP-hardorco-NP-hardfor}

H -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-hard

orco-NP-hardforH -freegraphs.

(vi) IfH

⊆

iP4,then ContractionBlocker

₍

χ

₎

ispolynomial-timesolvableforH -freegraphs,otherwiseitis NP-hardorco-NP-hard

forH -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-Contraction}Blocker(

ω

) _{is NP-complete} evenforC4-freeperfectgraphswith

ω

=

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-free

insteadof

(

H1

)

-free.Thecomplement ofG isthegraphG

= (

V

,

E

)

withvertexset V andanedgebetweentwoverticesu

andv 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 and

replacesitbyanewvertexw 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 K1,3is

the4-vertexstar,thatis,thegraphwithverticesu, v1,v2, v3andedgesuv1,uv2,uv3.

LetG

= (

V

,

E

)

bea graph.Asubset K

⊆

V iscalledaclique of G ifanytwo verticesin K areadjacenttoeach other. Thecliquenumber

ω

(

G

)

isthe numberofverticesinamaximumclique ofG.Asubset I

⊆

V iscalledanindependentset

of G ifanytwoverticesin I are non-adjacent toeach other.Theindependencenumber

α

(

G

)

isthenumberofverticesin amaximumindependentsetofG.Forapositiveintegerk, ak-colouring of G isamappingc

:

V

→ {

1

,

2

,

. . . ,

k

}

suchthat

c

(

u

)

=

c

(

v

)

wheneveruv

∈

E.Thechromaticnumber

χ

(

G

)

isthesmallestintegerk forwhichG hasak-colouring.Asubset ofedges M

⊆

E iscalledamatching ifno twoedgesof M shareacommonend-vertex.Thematchingnumber

μ

(

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 IndependentSet_{arethoseoftestingifagraphhasacliqueorindependentset,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.Agraphiscobipartite

ifitisthecomplementofabipartite (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-ClubContraction_{problem.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

and

π

=

ω

.Forournextresult(Theorem5) weneedsomeadditionalterminology.Abiclique

isacompletebipartitegraph,whichisnontrivial ifithasatleastoneedge.Abicliquevertex-partition ofagraphG

= (

V

,

E

)

isa set

S

ofmutuallyvertex-disjoint bicliquesin G suchthatevery vertexofG iscontainedinoneofthebicliquesof

S

. The BicliqueVertex-Partition_{problemconsistsintestingwhetheragivengraphG 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-timesolvableforbipartitegraphsand

k

=

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

)

-contractedintoagraphGwith

χ

(

G

)

≤

3 (sod

=

χ

(

G

)

−

3).

FirstsupposethatG hasabicliquevertex-partition

S

ofsize3.LetS1

,

S2

,

S3bethethree(nontrivial)bicliquesin

S

.Let Ai

,

Bi bethe twobipartitionclassesof Si fori

=

1

,

2

,

3.So,in G,we havethat A1

,

A2

,

A3, B1

,

B2

,

B3 are sixcliques that

partitionthe verticesof G,andmoreover,thereisnoedgebetweenavertexof Ai anda vertexof Bi,fori

=

1

,

2

,

3.In G

wecontract 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 with

χ

(

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 andbi

are 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 ai

corresponds to a setofvertices Ai

⊂

X (that we contractedinto thesingle vertexai) andthat each bi corresponds toa

setofvertices Bi

⊂

Y (thatwecontractedintothesingle vertexbi).Aseach pairai

,

bi isnon-adjacent,novertexof Ai is

adjacentto 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-titegraphsfor

π

∈ {

χ

,

ω

}.

For

π

=

χ

,wecanprovethisevenfortheclassofgraphswithindependencenumberatmost 2, orequivalently,theclassof3P1-freegraphs,whichproperlycontainstheclassofcobipartitegraphs.

Theorem6.Foranyfixedd

≥

0,the d-ContractionBlocker(

χ

)_{problemcanbesolvedinpolynomialtimefor3P1-freegraphs.}

Proof. LetG beagraphwith

α

(

G

)

≤

2.Consideracolouringwith

χ

(

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 whether

χ

(

G

)

≤

χ

(

G

)

−

d.Ifso,thenthealgorithmreturnsa yes-answer.Otherwise,thatis,if

χ

(

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

)_{problemcanbesolvedinpolynomialtimeon}

cobipartitegraphs.

Proof. For

π

=

χ

thisfollowsimmediatelyfromTheorem6.Ascobipartitegraphsareperfectandclosedunderedge con-traction,weobtainthesameresultfor

π

=

ω

.

2

Wenowconsidertheclassofbipartitegraphs.If

π

∈ {

χ

,

ω

}

,then ContractionBlocker(

π

)_{istrivialforbipartitegraphs} (andthus alsofortrees).Tothe contrary,for

π

=

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 sequenceof}

k

+

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 thisresultwemakeaconnectiontothematchingnumber

μ

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 results

inanewtreeT.Moreover, Thasn

−

1 verticesandtheedgecontractionneitherincreasedtheindependencenumbernor thematchingnumber.As

α

(

T

)

+

μ

(

T

)

=

n andsimilarly

α

(

T

)

+

μ

(

T

)

=

n

−

1,thismeansthateither

α

(

T

)

=

α

(

T

)

−

1 or

μ

(

T

)

=

μ

(

T

)

−

1.

Firstsupposethatd

≤

n

−

2

μ

(

T

)

.Thereareexactly

σ

(

T

)

=

n

−

2

μ

(

T

)

verticesthatareunsaturatedbyM. Letuv bean edge,suchthatu isunsaturated.AsM ismaximum,v mustbesaturated.Then,bycontractinguv,weobtainatreeTsuch that

μ

(

T

)

=

μ

(

T

)

.Itfollowsfromtheabovethat

α

(

T

)

=

α

(

T

)

−

1.Saythatwecontractedu ontov.TheninT wehave that v issaturatedby M,whichisamaximummatchingofT aswell.Thus,ifd

≤

n

−

2

μ

(

T

)

,contractingd edges,oneof the end-vertices ofwhichis unsaturatedby M, yields atree T with

μ

(

T

)

=

μ

(

T

)

and

α

(

T

)

=

α

(

T

)

−

d.Since an edge contractionreducestheindependencenumberbyatmost 1,itfollowsthatthisisoptimal.Hence,asd

≤

n

−

2

μ

(

T

)

,wefind that

(

G

,

T

,

k

)

isayes-instanceifk

≥

d andano-instanceifk

<

d.

Now supposethatd

>

n

−

2

μ

(

T

)

.Supposethatwefirstcontractthen

−

2

μ

(

T

)

edgesthathaveexactlyoneend-vertex that isunsaturatedby M.Itfollowsfromtheabove thatthisyields atreeT with

μ

(

T

)

=

μ

(

T

)

and

α

(

T

)

=

α

(

T

)

− (

n

−

2

μ

(

T

))

.SinceT doesnotcontainanyunsaturatedvertex, M isaperfectmatchingofT.Then,contractinganyedgeinT

resultsinatree T with

μ

(

T

)

=

μ

(

T

)

−

1 andthus,

α

(

T

)

=

α

(

T

)

.Ifwe contractanedge uv

∈

M,theresultingvertex

uv isunsaturatedby M

=

M

\ {

uv

}

in T.Hence, asexplainedabove,ifinadditionwe contractnowanedge

(

uv

)

w,we obtainatreeT with

α

(

T

)

=

α

(

T

)

−

1 and

μ

(

T

)

=

μ

(

T

)

.Repeatingthisprocedure,wemayreducetheindependence number of T byd with n

−

2

μ

(

T

)

+

2

(

d

−

n

+

2

μ

(

T

))

=

2

(

d

+

μ

(

T

))

−

n edge contractions.Below we show that thisis optimal.

Supposethatwecontractp edgesinT .LetTbetheresultingtree.Wehave

α

(

T

)

+

μ

(

T

)

=

n

−

p.As

μ

(

T

)

≤

1₂

(

n

−

p

)

, thismeansthat

α

(

T

)

≥

1₂

(

n

−

p

)

.Ifp

<

2

(

d

+

μ

(

T

))

−

n wehave

−

p₂

>

−(

d

+

μ

(

T

))

+

n₂,andthus

α

(

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

) when

π

∈ {

χ

_,

ω

}

.Inordertodosoweneedthefollowingrelation between 1-DeletionBlocker(

ω

)and VertexCover.

Proposition1.LetG beagraphwithatleastoneedgeandletk

≥

1 beaninteger.Then

(

G

,

ω

(

G

)

−

1

,

k

)

isayes-instanceof Deletion

Proof. 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 meansthat

ω

(

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 suchthat}

ω

₍

_G

−

_V

₎

≤

_{1.ThisimpliesthatG}

−

_V_hasnoedges. 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 of

1-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 where}

π

=

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 cobipartitegraphsif

π

∈ {

ω

,

χ

}

[13].Henceweonlyhavetodealwiththecase

π

=

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(see

forexample [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 denotethistreebyT_G.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 isacographandconstructingT_G (ifitexists)canbedonein timeO

(

n

+

m

)

.

Fortwointegersk and



wesaythatagraphG canbe

(

k

,

)

-contracted intoagraphH ifG canbemodifiedintoH bya sequencecontainingk edgecontractionsand



vertexdeletions.Notethatcographsareclosedunderedgecontractionand undervertexdeletion.Infact,toproveourresultsforcographs,wewillprovethefollowingmoregeneralresult.

Theorem11.Let

π

∈ {

α

,

χ

,

ω

}

.Theproblemofdeterminingthelargestinteger d suchthatacographG withn verticesandm edges canbe

(

k

,

)

-contractedintoacographH with

π

(

H

)

≤

π

(

G

)

−

d canbesolvedinO

(

n2

+

mn

+ (

k

+ )

3n

)

time.

Proof. Firstconsider

π

=

α

.LetG beacographwithn verticesandm edgesandletk

,



betwopositiveintegers.Wefirst constructT_G.WethenconsidereachnodeofT_G byfollowingabottom-upapproachstartingattheleavesofT_G andending initsrootr.

Let x be a node of T_G. Recall that Gx is the subgraph ofG induced by all vertices that corresponds to leavesin the

subtree of T_G 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 with

α

(

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,this

resultsintheemptygraph.Hence,d

(

i

,

j

,

x

)

=

α

(

Gx

)

.Fromnowonassumethati

+

j

<

|

V

(

Gx

)

|

or j

=

0.Assuch,anygraph

we 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

)

}

.In

ordertodetermined

(

i

,

j

,

x

)

wemustdosomefurtheranalysis.LetS beasequencethatconsistsofi edgecontractionsand

j vertexdeletionsofGx such thatapplying S onGx resultsina graphHx with

α

(

Hx

)

=

α

(

Gx

)

−

d

(

i

,

j

,

x

)

.We partition S

into five sets Se_y, Se_z, Se_yz, Sv_y, S_zv, respectively, asfollows. Let Se_y and Se_z 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 Szv

be thesetofdeletionsofverticesinGy andGz,respectively.Letb

= |

Svy

|

.Then

|

Svz

|

=

j

−

b.Wedistinguishbetweentwo

cases.

Firstassumethat Se

yz

= ∅

.Thenay

+

az

=

i.LetHybethegraphobtainedfromGy afterapplyingthesubsequenceofS,

consistingofoperationsinSe

y

∪

Svy,onGy.LetHzbedefinedanalogously.Thenwehave

α

(

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 Se_yz

= ∅

.Recallthati

+

j

<

|

V

(

Gx

)

|

or j

=

0.Hence

α

(

Hx

)

≥

1.Ourapproachisbasedonthefollowing

observations.

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

,

v

from 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.