Editing to a planar graph of given degrees

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Contents lists available atScienceDirect

Journal

of

Computer

and

System

Sciences

www.elsevier.com/locate/jcss

Editing

to

a

planar

graph

of

given

degrees

✩

Konrad

K. Dabrowski

a

,

∗

, Petr

A. Golovach

b

,

Pim van ’t

Hof

c

,

Daniël Paulusma

a

,

Dimitrios

M. Thilikos

d

,

e

,

f

a_School_of_Engineering_and_Computing_Sciences,_Durham_University,_United_Kingdom b_Department_of_Informatics,_University_of_Bergen,_Norway

c_School_of_Built_Environment,_Rotterdam_University_of_Applied_Sciences,_Rotterdam,_The_Netherlands d_Computer_Technology_Institute_and_Press_{“Diophantus”,}_Patras,_Greece

e_Department_of_Mathematics,_National_and_Kapodistrian_University_of_Athens,_Athens,_Greece f_AlGCo_{project-team,}_CNRS,_LIRMM,_Montpellier,_France

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received21February2016

Receivedinrevisedform 17November2016 Accepted26November2016

Availableonline1December2016 Keywords:

Graphediting Connectedgraph Planargraph Polynomialkernel

Weconsider thefollowinggraphmodiﬁcationproblem. Lettheinput consistofagraph

G_{= (}V,E),aweightfunctionw_:V_∪E_{→ N},acostfunctionc_:V_∪E_{→ N}0andadegree

functionδ:V→ N0,togetherwiththreeintegerskv,keand C .Thequestioniswhetherwe

candeleteasetofverticesoftotalweightatmost kv andasetofedgesoftotalweight

atmost ke sothat the totalcostofthe deleted elements isatmost C andevery

non-deletedvertex v hasdegreeδ(v)intheresultinggraph G.Wealsoconsiderthevariantin which G mustbeconnected.BothproblemsareknowntobeNP-completeandW[1]-hard whenparameterizedbykv+ke.Weprovethat,whenrestrictedtoplanargraphs,theystay

NP-completebuthavepolynomialkernelswhenparameterizedbykv+ke.

1. Introduction

Graphmodiﬁcationproblemscaptureavarietyoffundamentalgraph-theoreticproblems,andassuchtheyareverywell studied inalgorithmic graphtheory.The aimis tomodify some givengraph G intosome other graph H ,that satisﬁes a

certainproperty,byapplyingatmostsomegivennumberoperationsfromaset S ofprespeciﬁedgraphoperations.Well-known graph operations are the edge addition, edge deletion and vertex deletion, denoted by ea

,

ed and vd, respectively. For example,ifS

= {

vd

}

and H mustbeacliqueorindependentset,thenweobtaintwobasicgraphproblems,namely Clique and IndependentSet_,_{respectively.}_To_give_a_few_more_examples,_{if H must}_be_a_forest_and_either_S

= {

_ed

}

_or_S

= {

_vd

}

_,_then we obtain theproblems FeedbackEdgeSet_{and Feedback}VertexSet_,_{respectively.} _As _we_discuss_in _detail_later, _it _is_also commontoconsidersets S consistingofmorethanonegraphoperation.

✩ _An _extended_abstract_of_this_paper _appeared_in_the _proceedings_of_CSR₂₀₁₅_[10]._The_ﬁrst_and _fourth_{authors were} _supported_by_EPSRC_Grant

EP/K025090/1.TheresearchofthesecondauthorreceivedfundingfromtheEuropeanResearchCouncilundertheEuropeanUnion’sSeventhFramework Programme(FP/2007–2013)/ERCGrantAgreementn.267959.Theresearchoftheﬁfthauthorwasco-ﬁnancedbytheEuropeanUnion(EuropeanSocial FundESF)andGreeknationalfundsthroughtheOperationalProgram“EducationandLifelongLearning”oftheNationalStrategicReferenceFramework (NSRF)–ResearchFundingProgram:ARISTEIAII.

*

Correspondingauthor.

E-mailaddresses:konrad.dabrowski@durham.ac.uk(K.K. Dabrowski),petr.golovach@ii.uib.no(P.A. Golovach),p.van.t.hof@hr.nl(P. van ’tHof), daniel.paulusma@durham.ac.uk(D. Paulusma),sedthilk@thilikos.info(D.M. Thilikos).

http://dx.doi.org/10.1016/j.jcss.2016.11.009

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A property ishereditary if it isclosed under takinginduced subgraphs. A property isnon-trivial if itis both true for inﬁnitelymanygraphsandfalseforinﬁnitelymanygraphs.AclassicresultofLewisandYannakakis[24]isthatthevertex deletionproblemis NP-hard foranyproperty thatisboth hereditaryandnon-trivial. Inan earlierpaper,Yannakakis[33]

alsoshowedNP-hardnessresultsfortheedgedeletionproblemforseveralproperties,suchasbeingplanarorouter-planar.

Natanzon, Shamir andSharan [29] and Burzyn, Bonomo and Durán [5] proved that the graph modiﬁcation problem is

NP-completeforseveraltargethereditarygraphpropertieswhenS

= {

ea

,

ed

}

.

Aswecanseefromtheaboveresults,graphmodiﬁcationproblemsareoftenintractableevenforelementarycaseswhen

S

⊆ {

ea

,

ed

}

.Assuch,manypapersinthisarea studythecomplexity ofgraphmodiﬁcationproblemswhenparameterized bythetotalnumberofpermittedoperations k.

Cai[6]provedthatthegraphmodiﬁcationproblemisFPTwhenparameterizedby k,if S

= {

ea

,

ed

,

vd

}

andthedesired property isthat ofbelonging toanyﬁxed graph classcharacterized by a ﬁniteset offorbiddeninduced subgraphs. Khot andRaman[21] determinedallnon-trivialhereditarypropertiesforwhichthevertexdeletionproblemisFPTonn-vertex

graphswithparametern

−

k and proved thattheproblemis W

[

1

]

-hardwithrespectto thisparameter forallother such properties.

Fromtheaforementionedresultsweseethatthegraphmodiﬁcationproblemhasbeenthoroughlystudiedforhereditary properties.Severalothernaturaltypesofpropertieshavealsobeenconsidered.Forinstance,Dabrowskietal.[9]combined anumberofpreviousresults[4,7,8]withnewresultstogiveacompleteclassiﬁcationofthe(parameterized)complexityof theproblemofmodifying aninputgraphintoaconnectedgraphwhereeachvertexhassomeprescribeddegreeparityfor everysetS

⊆ {

ea

,

ed

,

vd

}

.

1.1. Ourfocus

In this paper we consider the case when the vertices of the resulting graph must satisfy some prespeciﬁed degree constraints (note that such properties are non-hereditary, so the resultof Lewis and Yannakakis does not apply to this case).Thisisa naturaldirectiontoconsidergiventhe classicalstructuralresults[25,32] onso-called f -factorsingraphs, whicharespanningsubgraphsinwhicheachvertex u musthavedegree f

(

u

)

forsomespeciﬁedfunction f (theseresults immediately imply that an f -factor in a graph can be found in polynomial time if one exists, while ﬁnding connected

f -factors,e.g.Hamiltoncycles,isNP-complete).

Beforepresentingourresults,webrieﬂydiscusstheknownresultsandthegeneralframeworktheyfallunder.

Generalframework. MoserandThilikosin[28] andMathiesonandSzeider[27] initiatedaninvestigationintothe param-eterizedcomplexityofgraphmodiﬁcationproblemswithrespecttodegreeconstraints.Thisleadstothefollowinggeneral problem.

DegreeConstraintEditing(S)

Instance: Agraph G,integersd,k andafunction

δ

:

V

(

G

)

→ {

1

,

. . . ,

d

}

.

Question: Can G bemodiﬁed intoagraph G suchthatdG

(

v

)

= δ(

v

)

foreach v

∈

V

(

G

)

usingatmost k operations

fromtheset S?

MathiesonandSzeider[27] classiﬁedtheparameterizedcomplexityofthisproblemfor S

⊆ {

ea

,

ed

,

vd

}

.Inparticularthey showedthefollowing results.If S

⊆ {

ea

,

ed

}

thenthe problemispolynomial-timesolvable. Ifvd

∈

S thenthe problemis NP-complete,W

[

1

]

-hardwithparameter k andFPTwithparameter d

+

k.Moreover,theyprovedthatthelatterresultholds evenforamoregeneralversion,inwhichtheverticesandedgeshavecostsandthedesireddegreeforeachvertexshould be insome givensubset of

{

1

,

. . . ,

d

}

. If

{

v

}

⊆

S

⊆ {

ed

,

vd

}

, they proved thatthe problemhasa polynomial kernelwhen parameterizedby d

+

k evenifverticesandedges havecosts. Recently,Mathieson[26] consideredgrapheditingproblems foranumberofalternativeformsofdegreeconstraints.Golovach[19]consideredthecasesS

= {

ea

,

vd

}

andS

= {

ea

,

ed

,

vd

}

andproved(amongstotherresults)thatforthesecasestheproblemhasnopolynomialkernelwhenparameterizedbyd

+

k

unless NP

⊆

coNP

/

poly. Froese,Nichterlein andNiedermeier [14] gave morekernelization resultsfor DegreeConstraint Editing(S).

Golovach[18] introduced avariantof DegreeConstraintEditing(S) withthe extracondition that theresulting graph must be connected. He proved that, for S

= {

ea

}

,this variantis NP-complete, FPTwhen parameterized by k, and hasa polynomialkernelwhenparameterizedby d

+

k. Theconnectedvariantisreadilyseen tobeW

[

1

]

-hard whenvd

∈

S bya straightforward modiﬁcationofthe proof ofthe W

[

1

]

-hardness resultfor DegreeConstraintEditing(S),when vd

∈

S,as givenbyMathiesonandSzeider[27].

Ourresults. InthelightoftheaboveNP-completenessandW

[

1

]

-hardness resultswhen vd

∈

S itisnaturaltorestrictthe inputgraph G toaspecialgraphclass.Hence,inspiredbytheaboveresults,weconsiderthesetS

= {

ed

,

vd

}

andstudyboth variantsoftheseproblems(whereweinsistthattheresultinggraph Gisconnectedandwherewedonot)forplanar input

graphs.Theproblemvariantnotdemandingconnectivityisdeﬁnedasfollows.(Infacttheproblemswe studyareslightly moregeneral.)

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PlanarDegreeConstraintDeletion

Instance: AplanargraphG

= (

V

,

E

)

,integerskv,keandafunction

δ

:

V

→ N

0.

Question: Can G bemodiﬁed into a graph G such that dG

(

v

)

= δ(

v

)

for each v

∈

V

(

G

)

usingat most kv vertex

deletionsandatmost ke edgedeletions?

We note that PlanarDegreeConstraintDeletion is NP-complete even if

δ

≡

3 and that its connected variant is

NP-complete evenif

δ

≡

2. Theseobservations followdirectlyfrom therespective facts that both testingwhethera pla-nargraphofmaximumdegreeatmost 7 hasanon-trivialcubicsubgraphisNP-complete[31]andtestingwhetheracubic planargraphhasaHamiltoniancycleisNP-complete[15].

IncontrasttotheaforementionedW

[

1

]

-hardnessresultsforgeneralgraphs,ourtwomainresultsarethattheweighted versionof PlanarDegreeConstraintDeletionanditsconnectedvariantbothhavepolynomialkernelswhenparameterized by kv

+

ke (see Section2.2fortheexact deﬁnitionoftheseweightedversions). Notethatby settingkv

=

0 orke

=

0 we

obtainthesameresultsfor DegreeConstrainedEditing(S) when S

= {

ed

}

andS

= {

vd

}

,respectively(thoughthe S

= {

ed

}

caseisnotsurprising,sincethisproblemissolvableinpolynomialtimeongeneralgraphs[27]).

In order to obtain our results we first show that both problemsare polynomial-timesolvable forany graph class of bounded treewidth.We thenuse avariantof theprotrusiondecomposition/replacement techniquesintroduced by Bodlaen-der et al.[3].Thesetechniqueswere successfullyusedforvariousproblemsonsparsegraphs[13,16,17,22].Westressthat our problems do not fit into the meta-algorithmic framework of Bodlaender et al. [3]on kernelization. Our kernels re-quireprotrusionreplacementmachinerythatisdifferentfromthegeneralone in[3].Hence ourapproachis,unavoidably, problem-specific.

2. Preliminaries

Inthissectionwestateterminologyandnotationusedthroughoutthepaper.

2.1. Basicterminologyandnotation

Allgraphsinthispaperareﬁnite,undirectedandwithoutloopsormultipleedges.Thevertexsetofagraph G isdenoted by V

(

G

)

andtheedgesetisdenotedby E

(

G

)

.ForasetX

⊆

V

(

G

)

,welet G

[

X

]

denotethesubgraphof G inducedby X .We let G

−

X

=

G

[

V

(

G

)

\

X

]

; notethat weallowthecasewhere X

V

(

G

)

.If X

= {

x

}

,wemaywrite G

−

x instead.Foraset

L

⊆

E

(

G

)

,weletG

−

L bethegraphobtainedfrom G bydeletingalledgesof L.IfL

= {

e

}

thenwemaywriteG

−

e instead.

For v

∈

V

(

G

)

,let EG

(

v

)

= {

e

∈

E

(

G

)

|

e is incident to v

}

.For X

⊆

V

(

G

)

,let EG

(

X

)

=

v∈XEG

(

v

)

.Fore

∈

E

(

G

)

withe

=

uv,

let V

(

e

)

= {

u

,

v

}

.ForasetL

⊆

E

(

G

)

let V

(

L

)

=

e∈LV

(

e

)

.

Let G be a graph.For a vertex v, we let NG

(

v

)

denote its (open)neighbourhood, that is,the set of vertices adjacent

to v. The degree ofa vertex v is denotedby dG

(

v

)

= |

NG

(

v

)

|

.Fora set X

⊆

V

(

G

)

,we write NG

(

X

)

= (

v∈XNG

(

v

))

\

X .

The closedneighbourhood NG

[

v

]

=

NG

(

v

)

∪ {

v

}

, andfora non-negative integer r, NrG

[

v

]

is the set ofvertices atdistance

at most r from v;note that N0

G

[

v

]

= {

v

}

andthat N1G

[

v

]

=

NG

[

v

]

.Let X

⊆

V

(

G

)

,andlet r be a positive integer. We let

Nr G

[

X

]

=

v∈XNrG

[

v

]

.The set X isan r-dominating set of G if V

(

G

)

⊆

NrG

[

X

]

.We let

∂

G

(

X

)

=

X

∩

NG

(

V

(

G

)

\

X

)

be the

boundary of X in G,i.e.thesetofverticesin X thathaveneighbours G outsideof X .

A treedecomposition ofa graph G isapair

(X ,

T

)

where T is atreeand

X = {

Xi

|

i

∈

V

(

T

)

}

isacollectionofsubsets

(calledbags)of V

(

G

)

suchthat

(i)

_i_∈_V₍_T₎Xi

=

V

(

G

)

,

(ii) foreachedgexy

∈

E

(

G

)

,thereisani

∈

V

(

T

)

suchthatx

,

y

∈

Xi,and

(iii) foreachx

∈

V

(

G

)

,theset

{

i

|

x

∈

Xi

}

inducesaconnectedsubtreeof T .

Thewidth ofatreedecomposition

(

{

Xi

|

i

∈

V

(

T

)

},

T

)

ismaxi∈V(T)

{|

Xi

|

−

1

}

.Thetreewidth ofagraph G (denoted tw

(

G

)

)is

the minimumwidthoverall treedecompositionsof G.Atreedecomposition

(X ,

T

)

ofagraph G is nice,if T isarooted binarytreesuchthatthenodesof T areoffourtypes:

(i) aleafnode i isaleafof T with Xi

= ∅

;

(ii) anintroducenode i hasonechild iwith Xi

=

Xi

∪ {

v

}

forsomevertexv

∈

V

(

G

)

;

(iii) aforgetnode i hasonechild iwith Xi

=

Xi

\ {

v

}

forsomevertexv

∈

VG;and

(iv) ajoinnode i hastwochildren iand i with Xi

=

Xi

=

Xi,

and, moreover,theroot r isaforgetnodewith Xr

= ∅

.Kloks[23]provedthateverytreedecompositionofagraphcanbe

convertedinlineartimetoanicetreedecompositionofthesamewidthsuchthatthesizeoftheobtainedtreeislinearin thesizeoftheoriginaltree.

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Lemma1.Let V1 and V2bethebipartitionclassesofaplanarbipartitegraph G suchthatdG

(

v

)

≥

3 forevery v

∈

V2and V2is

non-empty.Then

|

V2

|

≤

2

|

V1

|

−

4.

Proof. Let G besuchagraph.Let C

(

G

)

and F

(

G

)

bethesetofcomponentsandfacesin G,respectively.Since G isbipartite, theborderofeveryinternalfaceof G mustcontainatleastfouredges.Thisalsoappliestotheinﬁniteouterface,since G containsavertexofdegreeatleast 2 (edgescontainedintheborderofafacethatarenotpartofacyclearecountedtwice). Every edge is partof atmost two faces. It followsthat 4

|

F

(

G

)

|

≤

2

|

E

(

G

)

|

. Euler’sFormula forplanar graphs states that

|

V

(

G

)

|

−|

E

(

G

)

|

+|

F

(

G

)

|

−|

C

(

G

)

|

=

1.Combiningthiswiththeaboveinequality,weﬁndthat

|

E

(

G

)

|

≤

2

|

V

(

G

)

|

−

2

|

C

(

G

)

|

−

2

≤

2

|

V

(

G

)

|

−

4.

Now3

|

V2

|

≤

v∈V2dG

(

v

)

,sinceeveryvertexin V2 hasdegreeatleast 3.Weknowthat

v∈V2dG

(

v

)

= |

E

(

G

)

|

since G isbipartite.Combiningtheseobservationswiththeinequalityfoundaboveimpliesthat3

|

V2

|

≤

2

(

|

V1

|

+ |

V2

|)

−

4.Therefore

|

V2

|

≤

2

|

V1

|

−

4.

2

2.2. Fullproblemdescription

As mentionedin Section 1the problemswe solveare moregeneral than PlanarDegreeConstraintDeletionandits connectedvariant.The generalizationswe studyare analogousto thoseused forother editingproblems inthe literature (seee.g.[27]).Theunconnectedvariantthatwesolveisdeﬁnedasfollows:

DeletiontoaPlanarGraphofGivenDegrees(DPGGD)

Instance: AplanargraphG

= (

V

,

E

)

,integerskv,ke,C andfunctions

δ

:

V

→ N

0, w

:

V

∪

E

→ N

,c

:

V

∪

E

→ N

0.

Question: Can G bemodiﬁedintoagraph GbydeletingasetU

⊆

V withw

(

U

)

≤

kvandasetD

⊆

E withw

(

D

)

≤

ke

suchthatc

(

U

∪

D

)

≤

C anddG

(

v

)

= δ(

v

)

forv

∈

V

(

G

)

?

Inthisproblem, w istheweight and c isthecost function.Thequestioniswhetheritispossibletodeleteverticesandedges oftotalweightatmost kv and ke,respectively,sothatthetotalcostofthedeletedelementsisatmost C andtheobtained

graphsatisﬁesthedegreerestrictionsprescribedbythegivenfunction

δ

.Notethatifwedeleteavertex,theedgesincident to that vertexcan no longerbe present, so they are automatically deleted; theseedges do not contribute to theweight orcost ofthesolution.Weincludecosts to makeourresultsasgeneralaspossible.Inparticularnote that theinteger C isneithera constantnor aparameter,butpartofthe input.Addingcosts inthiswaydoesnotfundamentally complicate ourproof.Asthegoalistominimizethetotalcosts,thecostsforedgesandverticesarecombined.Besidescosts,wealso includeweights,mainlyfortechnicalreasons.

WecallthevariantofDPGGD,inwhichthedesiredgraph G mustbe connected,the DeletiontoaConnectedPlanar

GraphofGivenDegrees_problem_(DCPGGD).

2.3. Protrusiondecompositions

For a graph G anda positive integer r, a set X

⊆

V

(

G

)

is an r-protrusion of G if

|∂

G

(

X

)

|

≤

r and tw(G

[

X

])

≤

r. For

positiveintegers s and s,an

(

s

,

s

)

-protrusiondecompositionofagraph G (seealsoFig. 1) isa partition

= {

R0

,

. . . ,

Rp

}

of V

(

G

)

suchthat

(i) max

{

p

,

|

R0

|}

≤

s,

(ii) foreachi

∈ {

1

,

. . . ,

p

}

,R+_i

=

NG

[

Ri

]

isans-protrusionof G,and

(iii) foreachi

∈ {

1

,

. . . ,

p

}

,NG

(

Ri

)

⊆

R0

∩ ∂

G

[

R+i

]

.

The sets R+₁

,

. . . ,

R+_p are called the protrusions of

. Originally, condition (iii) only demanded that NG

(

Ri

)

⊆

R0 holds

foreach i

∈ {

1

,

. . . ,

p

}

.However, we can move every vertex in NG

(

Ri

)

\ ∂

G

[

R+i

]

to Ri without affectingany ofthe other

properties.Henceweassumewithoutlossofgeneralitythatsuchverticesdonotexistandmayindeedstatecondition(iii) asabove(whichisconvenientforourpurposes).Notethatifavertexv

∈

R_i+hasaneighbouroutsideof R+_i thenv

∈

NG

(

Ri

)

bythedeﬁnitionof R+_i.Itfollowsthateveryvertexof

∂

G

[

R+i

]

alsoliesinNG

(

Ri

)

andthereforeNG

(

Ri

)

= ∂

G

[

R+i

]

.

Thefollowingstatementisimplicitin[3](seeLemmas 6.1and 6.2).

Lemma2([3]).Let r and k bepositiveintegersandlet G beaplanargraphthathasanr-dominatingsetofsizeatmost k.Then G has an

(

O

(

kr

),

O

(

r

))

-protrusiondecomposition,whichcanbeconstructedinpolynomialtime.

2.4. Parameterizedcomplexity

Parameterizedcomplexityisatwodimensionalframeworkforstudyingthecomputationalcomplexityofaproblem.One dimension is theinput size n and the other is aparameter k. A problemis said tobe ﬁxedparametertractable (or FPT)

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Fig. 1. A protrusion decomposition with p=3. Recall that R+_i =NG[Ri]and note that the sets∂G(R+i)=NG(Ri)are not necessarily pairwise disjoint.

if itcan be solved intime f

(

k

)

·

nO(1) _for_some _{function f .}_{A kernelization for} _a _{parameterized}_problem_is _a_polynomial

algorithmthatmapseachinstance

(

x

,

k

)

withinput x andparameter k toaninstance

(

x

,

k

)

suchthat

(i)

(

x

,

k

)

isayes-instanceifandonlyif

(

x

,

k

)

isayes-instance,and (ii) thesizeof xand kisboundedby f

(

k

)

foracomputablefunction f .

The output

(

x

,

k

)

is calleda kernel. The function f is said to be the size of the kernel. A kernel is polynomial if f is

polynomial. We refer tothe booksofDowney andFellows [11],FlumandGrohe [12],andNiedermeier [30] fordetailed introductionstoparameterizedcomplexity.

3. Thepolynomialkernels

In this section we construct polynomial kernels for DPGGD and DCPGGD. We say that a pair

(

U

,

D

)

with U

⊆

V

(

G

)

and D

⊆

E

(

G

)

is a solution for an instance

(

G

,

kv

,

ke

,

C

,

δ,

w

,

c

)

of DPGGDif w

(

U

)

≤

kv, w

(

D

)

≤

ke, c

(

U

∪

D

)

≤

C and

G

=

G

−

U

−

D satisﬁesdG

(

v

)

= δ(

v

)

forall v

∈

V

(

G

)

.If

(

G

,

kv

,

ke

,

C

,

δ,

w

,

c

)

isaninstanceofDCPGGDthen

(

U

,

D

)

isa

solutionifinaddition Gisconnected.NoticethatitcanhappenthatU

=

V

(

G

)

forasolution

(

U

,

D

)

.

In ordertoproveour mainresults,weﬁrst needtointroduce someadditionalterminologyandprovesome structural results. We say that a solution

(

U

,

D

)

is ofminimumcost if c

( ˆ

U

,

D

ˆ

)

≥

c

(

U

,

D

)

for every solution

( ˆ

U

,

D

ˆ

)

.We say that a solution

(

U

,

D

)

foran instance of DPGGD or DCPGGD is eﬃcient if D has no edges incident tothe vertices of U . Since deleting a vertex automatically removes all incident edges with no weight or cost penalty,we can make the following observation.

Observation1.Anyyes-instanceof DPGGDor DCPGGDhasaneﬃcientsolutionofminimumcost.

Wewillalsomakeuseofthefollowingsimpleobservation.

Observation2.Let

(

G

,

kv

,

ke

,

C

,

δ,

w

,

c

)

beinstanceof DPGGDor DCPGGDthathasaneﬃcientsolution

(

U

,

D

)

.IfdG

(

v

)

= δ(

v

)

for

somev

∈

V

(

G

)

then v isnotincidenttoanedgeof D.

Wesaythataninstance

(

G

,

kv

,

ke

,

C

,

δ,

w

,

c

)

of DPGGD (DCPGGD respectively)isnormalized if

(i) forevery v

∈

V

(

G

)

,

δ(

v

)

≤

dG

(

v

)

≤ δ(

v

)

+

kv

+

ke,and

(ii) everyvertex v inthesetS

= {

u

∈

V

(

G

)

|

dG

(

u

)

= δ(

u

)

}

isadjacenttoavertexinS

=

V

(

G

)

\

S.

Lemma3.Thereisapolynomial-timealgorithmthatforeachinstanceof DPGGDor DCPGGDeithersolvestheproblemorreturnsan equivalentnormalizedinstance.

Proof. Let

(

G

,

kv

,

ke

,

C

,

δ,

w

,

c

)

beaninstanceof DPGGD.Tosimplifynotation,wekeepthesamenotationforthefunctions

δ,

w

,

c ifwedeleteverticesoredgesanddonotmodifythevaluesofthefunctionsfortheremainingelementsifthisdoes notcreateconfusion.

Wesaythatareductionruleissafe ifbyapplyingtheruleweeithersolvetheproblemorobtainanequivalentinstance. Itisstraightforwardtoseethatthefollowingreductionrulesaresafe.

(6)

Yes-instancerule.IfS

=

V

(

G

)

then

(

∅,

∅)

isasolution,returnayes-answerandstop.

Vertexdeletionrule.If G hasavertex v withdG

(

v

)

< δ(

v

)

ordG

(

v

)

> δ(

v

)

+

kv

+

ke,thendelete v andsetkv

=

kv

−

w

(

v

)

,

C

=

C

−

c

(

v

)

.Ifkv

<

0 orC

<

0,thenstopandreturnano-answer.

Observe that by the exhaustive application ofthe vertexdeletionrule and applyingthe yes-instancerule whenever

possible,weeithersolvetheproblemorweobtainaninstancewhichsatisﬁes(i)ofthedeﬁnitionofnormalizedinstances, butwhere S

=

V

(

G

)

.Notice that, inparticular, the yes-instancerule is appliedifthe set ofverticesbecomes empty.To ensure(ii),weapplythefollowingtworules.

Contractionrule.If G hastwoadjacentverticesu

,

v

∈

S

= {

x

∈

V

(

G

)

|

dG

(

x

)

= δ(

x

)

}

suchthatNG

(

v

)

⊆

S,thenweconstruct

theinstance

(

G

,

kv

,

ke

,

C

,

δ

,

w

,

c

)

asfollows.

– Contract uv.DenotetheobtainedgraphG

=

G

/

uv andlet z bethevertexobtainedfrom u and v. – Set

δ

(

z

)

=

dG

(

z

)

andset

δ

(

x

)

=

dG

(

x

)

foranyx

∈

S

\ {

u

,

v

}

.Foreachx

∈

S,set

δ

(

x

)

= δ(

x

)

.

– Setw

(

z

)

=

w

(

u

)

+

w

(

v

)

andc

(

z

)

=

c

(

u

)

+

c

(

v

)

.Forx

∈

V

(

G

)

\ {

u

,

v

}

,setw

(

x

)

=

w

(

x

)

andc

(

x

)

=

c

(

x

)

.

– Foreachxz

∈

E

(

G

)

,set w

(

xz

)

=

ke

+

1 andc

(

xz

)

=

0.Forallotheredges xy

∈

E

(

G

)

,set w

(

xy

)

=

w

(

xy

)

andc

(

xy

)

=

c

(

xy

)

.

Let

(

U

,

D

)

bean eﬃcientsolutionfor

(

G

,

kv

,

ke

,

C

,

δ,

w

,

c

)

.ByObservation 2, D has noedges incidentto u or v. Also

either u

,

v

∈

U or u

,

v

∈

/

U , because u and v areadjacentanddG

(

u

)

= δ(

u

)

anddG

(

v

)

= δ(

v

)

.LetU

= (

U

\ {

u

,

v

})

∪ {

z

}

ifu

,

v

∈

U andU

=

U otherwise.We havethat

(

U

,

D

)

isa solutionfor

(

G

,

kv

,

ke

,

C

,

δ

,

w

,

c

)

.If

(

U

,

D

)

is an eﬃcient

solutionfor

(

G

,

kv

,

ke

,

C

,

δ

,

w

,

c

)

,then Dhasnoedgesincidentto z byObservation 2.Ifz

∈

U,letU

= (

U

\ {

z

})

∪ {

u

,

v

}

andU

=

Uotherwise.Weobtainthat

(

U

,

D

)

isasolutionfortheoriginalinstance.

Weexhaustivelyapplytheaboverule.Assumethatitcannotbeappliedfor

(

G

,

kv

,

ke

,

C

,

δ,

w

,

c

)

.Thenwehavethatthis

instancesatisﬁes (i)andthefollowingholds:foranyv

∈

S

=

V

(

G

)

,either v isadjacenttoavertexin S or v isanisolated vertex.Itremainstodealwithisolatedvertices.

Isolatesremovalrule.If G hasanisolatedvertex v,thendelete v.

Toseethataboveruleissafe,noticethat,becausetheconsideredinstancesatisﬁes(i),itfollowsthat

δ(

v

)

≤

dG

(

v

)

=

0,so

v

∈

S.Clearly,bytheexhaustiveapplicationofthe isolatesremovalrule,weeithersolvetheproblemorobtainaninstance thatsatisﬁes(i)and(ii).

Nowconsideraninstance

(

G

,

kv

,

ke

,

C

,

δ,

w

,

c

)

of DCPGGD.

Wereplacethe yes-instancerule bythefollowingvariant.

Yes-instancerule(connected).IfS

=

V

(

G

)

and G isconnected,then

(

∅,

∅)

isasolution,returnayes-answerandstop. Itisstraightforwardtoverifythatthe vertexdeletionrule andthe contractionrule aresafeforthisproblem.Byapplying theserulesandbytheapplicationoftheconnectedvariantofthe yes-instancerule wheneverpossible,weeithersolvethe problemorobtain anequivalent instancethatsatisﬁes (i)andhasthepropertythat forany v

∈

S,either v isadjacentto a vertexin S or v isan isolated vertex.Suppose that

(

G

,

kv

,

ke

,

C

,

δ,

w

,

c

)

satisﬁestheseproperties.Observethat if H is

a componentof G, thenfor anysolution

(

U

,

D

)

, either V

(

H

)

⊆

U or V

(

G

)

\

V

(

H

)

⊆

U . Therefore,itis safetoapply the followingvariantofthe isolatesremovalrule.

Isolatesremovalrule(connected).If G has anisolated vertex v, thenif w

(

V

(

G

)

\ {

v

})

≤

kv andc

(

V

(

G

)

\ {

v

})

≤

C ,then

(

V

(

G

)

\ {

v

},

∅)

isasolution,returnayes-answerandstop.Otherwise,if w

(

V

(

G

)

\ {

v

})

>

kv orc

(

V

(

G

)

\ {

v

})

>

C ,delete v

andsetkv

=

kv

−

w

(

v

)

andC

=

C

−

c

(

v

)

;ifkv

<

0 orC

<

0,thenstopandreturnano-answer.

Itis easyto seethat iftheinput graphwas planarthen thegraphformed afterapplyingtherules abovewill alsobe

planar.

2

Lemma4.If

(

G

,

kv

,

ke

,

C

,

δ,

w

,

c

)

isanormalizedyes-instanceof DPGGD(DCPGGDrespectively)then G hasa2-dominatingsetof

sizeatmostkv

+

2ke.

Proof. We prove the lemma for DPGGD; the proof for DCPGGD is the same. Let

(

G

,

kv

,

ke

,

C

,

δ,

w

,

c

)

be a normalized

yes-instanceoftheproblem.Let

(

U

,

D

)

beasolutionandW

=

U

∪

V

(

D

)

.Clearly,

|

W

|

≤

kv

+

2ke,becausetheweightsare

(7)

Let S

= {

v

∈

V

(

G

)

|

dG

(

v

)

= δ(

v

)

}

andS

=

V

(

G

)

\

S.Foranyvertexv

∈

S,eitherv

∈

U or v isadjacenttoavertexof U

or v isincidenttoanedgeof D.Hence,S

⊆

NG

[

W

]

.Letv

∈

S.Becausetheconsideredinstanceisnormalized, v isadjacent

toavertexu

∈

S.Itimplies,that S

⊆

N2_G

[

W

]

.

2

ThefollowingisadirectconsequenceofLemmas 2 and 4.

Lemma5.Thereisaﬁxedconstant

α

suchthat,if

(

G

,

kv

,

ke

,

C

,

δ,

w

,

c

)

isanormalizedyes-instanceof DPGGD(DCPGGD

respec-tively),then G hasan

(

α

(

kv

+

2ke

),

α

)

-protrusiondecomposition.Moreover,ifthereissuchadecomposition,onecanbeconstructed

inpolynomialtime.

The next lemma statesthat, for both DPGGD and DCPGGD,an optimal solution can be found inpolynomial time on

graphsofboundedtreewidth.Theproofisbasedonthestandardtechniquesfordynamicprogrammingovertree decompo-sitions.

Lemma6. DPGGD(DCPGGD respectively)canbe solved,andan eﬃcientsolution

₍

U

_,

D

₎

ofminimumcostcanbeobtained in

(

kv

+

ke

)

O(q)

·

poly

(

n

)

time(in

(

q

(

kv

+

ke

))

O(q)

·

poly

(

n

)

timerespectively)forinstances

(

G

,

kv

,

ke

,

C

,

δ,

w

,

c

)

where G isann-vertex

graphoftreewidthatmost q and

δ(

v

)

≤

dG

(

v

)

≤ δ(

v

)

+

kv

+

keforv

∈

V

(

G

)

.

Proof. Weuseamoreorlessstandardapproachforconstructionofdynamicprogrammingalgorithmsforgraphsofbounded treewidth.

First,weconsider DPGGD. Let

(

G

,

kv

,

ke

,

C

,

δ,

w

,

c

)

beaninstanceoftheproblemwheretw(G

)

≤

q and

δ(

v

)

≤

dG

(

v

)

≤

δ(

v

)

+

kv

+

ke forall v

∈

V

(

G

)

.Weﬁrstofall assumethata nicetreedecomposition

(X ,

T

)

of G withwidtht

=

O

(

q

)

is

given. Tosimplifylaterarguments,wemayassumet

≥

2.Forthis,wemayusethealgorithmof[2]toobtaina decomposi-tionwhosewidthisatmostﬁvetimestheoptimalin2O(q)

_·

_{n steps}_and_then_convert_it_to_a_nice_tree_{decomposition}_using

theaforementionedresultsofKloks[23].

Let r denotetherootof T .Foranynode i

∈

V

(

T

)

,let Ti denotethesubtreeof T inducedby i anditsdescendantsand

let Gi

=

G

[

j∈V(Ti)Xj

]

.Weapplyadynamicprogrammingalgorithmover

(X ,

T

)

.

First,wedescribethetablesthatareconstructedforthenodesof T .Leti

∈

V

(

T

)

.Wedeﬁnetableiasapartialfunction

whoseinputsarequintuples

(

X

,

Y

,

γ

,

hv

,

he

)

where

– X

⊆

Xi,

– Y

⊆

E

(

G

[

Xi

])

,

–

γ

:

Xi

\

X

→ {

0

,

. . . ,

kv

+

ke

}

,

– hv

≤

kv and

– he

≤

ke.

Thevalueoftablei isaminimumcostpair

(

U

,

D

)

∈

2V(Gi)

×

2E(Gi)withthefollowingproperties:

(i) foranyv

∈

U andanye

∈

D,v and e arenotincident, (ii) w

(

U

)

≤

hv andw

(

D

)

≤

he,

(iii) U

∩

Xi

=

X andD

∩

E

(

G

[

Xi

])

=

Y ,

(iv) foreveryv

∈

Xi

\

X ,thenumberofneighboursof v in GithatbelonginU

\

XiplusthenumberofedgesofD

\

E

(

G

[

Xi

])

thatareincidentto v isexactly

γ

(

v

)

,

(v) foreach v

∈

V

(

Gi

)

\

Xi,dG_i

(

v

)

= δ(

v

)

whereGi

=

Gi

−

U

−

D,

and,ifnosuchpair

(

U

,

D

)

exists,thentablei

(

X

,

Y

,

d

,

hv

,

he

)

isvoid.

Recall that Xr

= ∅

. Observethat

(

G

,

kv

,

ke

,

C

,

δ,

w

,

c

)

is a yes-instanceifand onlyif tabler

(

∅,

kv

,

ke

)

is non-void

(where

∅

: ∅

→ {

0

,

. . . ,

kv

+

ke

}

).Moreover,insuchacase,thevalueoftabler

(

∅,

kv

,

ke

)

isaminimum-costsolutionfor

thisinstance.

Nowweexplainhowweconstructtableiforeachi

∈

V

(

T

)

.If i isaleaf node,tablei isconstructedinastraightforward

waybecause Xi

= ∅

.Indeed,for0

≤

hv

≤

kv and0

≤

he

≤

ke we settablei

(

∅,

hv

,

he

)

= (∅,

∅)

andhavetablei voidin

allothercases.Hence,itremainstogivetheconstructionforintroduce,forget,andjoin nodes.Leti

∈

V

(

T

)

beanodeofone ofthesetypes.Assumeinductivelythatthefunctiontablei foreverychild iof i hasalreadybeenconstructed.

In whatfollowswe write tablei

(

X

,

Y

,

γ

,

hv

,

he

)

(

U

,

D

)

to referto thefollowingprocedure: Iftablei

(

X

,

Y

,

γ

,

hv

,

he

)

is undeﬁned, set it to be equal to

(

U

,

D

)

. If tablei

(

X

,

Y

,

γ

,

hv

,

he

)

= ( ˆ

U

,

D

ˆ

)

and c

( ˆ

U

∪ ˆ

D

)

>

c

(

U

∪

D

)

, change tablei

(

X

,

Y

,

γ

,

hv

,

he

)

tobeequalto

(

U

,

D

)

.Otherwise,donotchangetablei

(

X

,

Y

,

γ

,

hv

,

he

)

.

Constructionforanintroducenode. Let i be thechild of i and Xi

=

Xi

∪ {

v

}

.Notice that NGi

(

v

)

⊆

Xi.We start with

tablei empty.Then, foreach pair hv

,

he where hv

≤

kv andhe

≤

ke and each pair

((

X

,

Y

,

γ

,

hv

,

he

),

(

U

,

D

))

∈

tablei

(8)

– LetX

←

X

∪ {

v

}

,Y

←

Y,

γ

←

γ

,U

←

U

∪ {

v

}

,and D

←

D. Ifhv

≥

hv

+

w

(

v

)

,thentablei

(

X

,

Y

,

γ

,

hv

,

he

)

(

U

,

D

)

.

– LetX

←

X,U

←

U,

γ

←

γ

∪ {(

v

,

0

)

}

.

For every L

⊆ {

vu

|

vu

∈

E

(

G

),

u

∈

Xi

\

X

}

, let Y

←

Y

∪

L, D

←

D

∪

L, and if he

≥

he

+

w

(

L

)

, then tablei

(

X

,

Y

,

γ

,

hv

,

he

)

(

U

,

D

)

.

Constructionforaforgetnode. Let i be the child of i and Xi

=

Xi

\ {

v

}

. We start with tablei empty. For each pair

((

X

,

Y

,

γ

,

hv

,

he

),

(

U

,

D

))

∈

tablei,wedothefollowing.

– If v

∈

X then let X

←

X

\ {

v

}

, Y

←

Y,anddeﬁne

γ

by replacingin

γ

each pair

(

u

,

γ

(

u

))

where uv

∈

E

(

G

)

and

u

∈

Xi

\

X bythepair

(

u

,

γ

(

u

)

+

1

)

.

Ifmaxu∈Xi\X

γ

(

u

)

≤

kv

+

ke,thentablei

(

X

,

Y

,

γ

,

hv

,

he

)

(

U

,

D

)

.

– Ifv

∈

/

X,thenlet X

←

X,L

← {

vu

∈

E

(

G

)

|

u

∈

Xi

}

∩

Y,Y

←

Y

\

L,anddeﬁne

γ

byreplacingin

γ

−

=

γ

\ {(

v

,

γ

(

v

))

}

eachpair

(

u

,

γ

(

u

))

whereuv

∈

L bythepair

(

u

,

γ

(

u

)

+

1

)

.

If

δ(

v

)

=

dG

(

v

)

− |

L

|

−

γ

(

v

)

andmaxu∈Xi\X

γ

(

u

)

≤

kv

+

ke,thentablei

(

X

,

Y

,

γ

,

hv

,

he

)

(

U

,

D

)

.

Constructionforajoinnode. Let iand ibethechildrenof i.Westartwithtableiempty.Foreachpair

((

X

,

Y

,

γ

,

hv

,

he

),

(

U

,

D

))

∈

tablei andeachpair

((

X

,

Y

,

γ

,

hv

,

he

),

(

U

,

D

))

∈

tablei wedothefollowing.

– Let

γ

←

γ

+

γ

,U

←

U

∪

UandD

←

D

∪

D.

Ifmaxu∈Xi\X

γ

(

u

)

≤

kv

+

ke,thenforanytwointegershv,he suchthathv

+

hv

−

w

(

X

)

≤

hv

≤

kv andhe

+

he

−

w

(

Y

)

≤

he

≤

ke,tablei

(

X

,

Y

,

γ

,

hv

,

he

)

(

U

,

D

)

.

Usingstandardarguments,itisstraightforwardtoverifythecorrectnessofthealgorithm.Toevaluatetherunningtime, recallthattablei receivesa quintuple

(

X

,

Y

,

γ

,

hv

,

he

)

asinput.There areatmost 2t+1 possiblechoicesfor X ,23(t+1)−6

=

23t−3 _choices_{of Y (because}_of_the_planarity_{of G),}

₍

_k

v

+

ke

+

1

)

t+1 choicesof

γ

,kv

+

1 possiblevaluesof hv andke

+

1

possiblevaluesfor he.Wethereforehavethateach tablei has

(

kv

+

ke

)

O(t) entries.Thisimpliesthat therunningtime of

thedynamicprogrammingalgorithmis

(

kv

+

ke

)

O(t)

·

n.

Nowweconsider DCPGGD.Thedifferenceisthatwehavetokeeptrackofcomponentsofapartialsolutionasisstandard fordynamicprogrammingalgorithmsforgraphsofboundedtreewidthwithaconnectivityconditionsuchas,e.g.the Steiner Treeproblem. Let

(

G

,

k_v

,

k_e

,

C

,

δ,

w

,

c

)

bean instanceof DCPGGD wheretw(G

)

≤

t and

δ(

v

)

≤

d_G

(

v

)

≤ δ(

v

)

+

k_v

+

k_e for

v

∈

V

(

G

)

.Without lossof generalitywe assume that a nicetreedecomposition

(

X ,

T

)

of G withtreewidth atmost t is givenandapplyadynamicprogrammingalgorithmover

(X ,

T

)

.Leti

∈

V

(

T

)

.

Wedeﬁnetablec_i asapartialfunctionwhoseinputsarequintuples

(P,

Y

,

γ

,

hv

,

he

)

where

–

P = {

P0

,

. . . ,

Ps

}

isapartitionof Xi,

– Y

⊆

E

(

G

[

Xi

])

,

–

γ

:

Xi

\

X

→ {

0

,

. . . ,

kv

+

ke

}

,

– hv

≤

kv and

– he

≤

ke.

Thevalueoftablec_i isaminimumcostpair

(

U

,

D

)

∈

2V(Gi)

×

₂E(Gi)_with_the_following_properties: (i) foranyv

∈

U andanye

∈

D, v and e arenotincident,

(ii) w

(

U

)

≤

hv andw

(

D

)

≤

he,

(iii) U

∩

Xi

=

P0 andD

∩

E

(

G

[

Xi

])

=

Y ,

(iv) foreveryv

∈

Xi

\

X ,thenumberofneighboursof v in GithatbelonginU

\

Xi,plusthenumberofedgesofD

\

E

(

G

[

Xi

])

thatareincidentto v isexactly

γ

(

v

)

,

(v) foreachv

∈

V

(

Gi

)

\

Xi,dG_i

(

v

)

= δ(

v

)

whereGi

=

Gi

−

U

−

D,

(vi) ifs

=

0,thenG_i

=

Gi

−

U

−

D isconnectedandifs

≥

1,then Gihas s componentsH1

,

. . . ,

HssuchthatV

(

Hi

)

∩

Xh

=

Pi

forh

∈ {

1

,

. . . ,

s

}

,

and,ifnosuchpair

(

U

,

D

)

exists,thentable_ic

(P,

Y

,

d

,

hv

,

he

)

isvoid.

Asinthenon-connectedcase,

(

G

,

kv

,

ke

,

C

,

δ,

w

,

c

)

isayes-instanceifandonlyiftablecr

(

∅,

kv

,

ke

)

isnon-voidand

thevalueoftablec_r

(

∅,

kv

,

ke

)

,ifexists,isaminimum-costsolutionforthisinstance.

Thepartialfunctiontablec_i isconstructedforeveryi

∈

V

(

T

)

similarlytotheconstruction oftablei for DPGGD.Because

thereareatmost

(

t

+

1

)

t+1partitions

P

ofeach Xi,wehavethateachtablecontains

(

t

(

kv

+

ke

))

O(t)entries.Therefore,the

runningtimeofthedynamicprogrammingalgorithmis

(

t

(

kv

+

ke

))

O(t)n.

2