Locally constrained homomorphisms on graphs of bounded treewidth and bounded degree

Contents lists available atScienceDirect

Theoretical

Computer

Science

www.elsevier.com/locate/tcs

Locally

constrained

homomorphisms

on

graphs

of

bounded

treewidth

and

bounded

degree

✩

Steven Chaplick

a

,

1

,

Jiˇrí Fiala

b

,

2

,

Pim van ’t Hof

c

,

Daniël Paulusma

d

,

∗

,

Marek Tesaˇr

b

a_{InstitutfürMathematik,TUBerlin,Germany}

b_{DepartmentofAppliedMathematics,CharlesUniversity,Prague,CzechRepublic} c_{DepartmentofInformatics,UniversityofBergen,Norway}

d_{SchoolofEngineeringandComputingSciences,DurhamUniversity,UK}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received 5 September 2013

Received in revised form 26 October 2014 Accepted 16 January 2015

Available online 21 January 2015 Keywords:

Computational complexity

Locally constrained graph homomorphisms Bounded treewidth

Bounded degree

AhomomorphismfromagraphG toagraphH islocallybijective,surjective,orinjective if its restriction to the neighborhood of every vertex of G is bijective, surjective, or injective,respectively. We prove thatthe problemsof testingwhether agivengraph G

allowsahomomorphismtoagivengraphH thatislocallybijective,surjective,orinjective, respectively,are NP-complete,evenwhenG haspathwidthatmost5,4,or2,respectively, orwhenbothG andH havemaximumdegree 3.Wecomplementthesehardnessresultsby showingthatthethreeproblemsarepolynomial-timesolvableifG hasboundedtreewidth andinadditionG orH hasboundedmaximumdegree.

©2015ElsevierB.V.All rights reserved.

1. Introduction

Allgraphsconsideredinthispaperarefinite,undirected,andhaveneitherself-loopsnormultipleedges.A

graph

homo-morphism from agraph

G

= (

VG

,

E

G

)

toagraph H

= (

VH

,

EH

)

isamapping

ϕ

:

VG

→

VH thatmapsadjacentverticesof

G

to adjacentverticesof H , i.e.,

ϕ

(

u

)

ϕ

(

v

)

∈

EH whenever

uv

∈

EG.Thenotionofagraphhomomorphismiswell studiedin

theliteratureduetoitsmanypracticalandtheoreticalapplications;werefertothetextbookofHellandNešetˇril[29]fora survey.

Wewrite

G

→

H to indicatetheexistenceofahomomorphismfrom

G to

H . Wecall

G the guest graph and H the host

graph. We denotetheverticesof H by 1

,

. . . ,

|

H

|

andcall them

colors.

The reasonfordoingthisisthat graph homomor-phismsgeneralizegraphcolorings:thereexistsahomomorphismfromagraph

G to

acompletegraphon

k vertices

ifand onlyif

G is k-colorable.

Theproblemoftestingwhether

G

→

H for twogivengraphs

G and H is

calledthe Hom problem.If onlytheguestgraphispartoftheinputandthehostgraphis

fixed,

i.e.,notpartoftheinput,thenthisproblemisdenoted

✩ _{This paper is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Research Council of Norway (197548/F20),} EPSRC (EP/G043434/1) and the Royal Society (JP100692). An extended abstract of it appeared in the Proceedings of FCT 2013[10].

*

Corresponding author.

E-mailaddresses:chaplick@math.tu-berlin.de(S. Chaplick), fiala@kam.mff.cuni.cz(J. Fiala), pim.vanthof@ii.uib.no(P. van ’t Hof), daniel.paulusma@durham.ac.uk(D. Paulusma), tesar@kam.mff.cuni.cz(M. Tesaˇr).

1 _{Supported by the ESF GraDR EUROGIGA grant as project GACR GIG/11/E023 and the NSERC grants of: K. Cameron and C. Hoàng (Wilfrid Laurier} University), D. Corneil (University of Toronto), and P. Hell (Simon Fraser University).

2 _{Supported by MŠMT ˇCR grant LH12095 and GAˇCR grant P202/12/G061.}

http://dx.doi.org/10.1016/j.tcs.2015.01.028 0304-3975/©2015 Elsevier B.V. All rights reserved.

as

H -Hom.

TheclassicalresultinthisareaistheHell–Nešetˇrildichotomytheoremwhichstatesthat H -Hom is solvablein polynomialtimeifH is bipartite,and NP-completeotherwise[27].

We consider so-called locally constrained homomorphisms. The neighborhood of a vertex u in a graph G is denoted

NG

(

u

)

= {

v

∈

VG

|

uv

∈

EG

}

. If for every u

∈

VG the restriction of

ϕ

to the neighborhood of u, i.e., the mapping

ϕ

u

:

NG

(

u

)

→

NH

(

ϕ

(

u

))

, is injective, bijective, or surjective, then

ϕ

is said to be locally injective, locally bijective, or lo-cally surjective, respectively.Locallybijectivehomomorphismsarealsocalled

graph coverings.

Theyoriginatefromtopological graph theory [4,37] and have applications in distributed computing [2,3,7] and in constructing highly transitive regular graphs [5]. Locally injective homomorphisms are also calledpartial graph coverings. Theyhave applications in models of telecommunication [16] andindistance constrainedlabeling [17].Moreover, they are used asindicatorsofthe existence ofhomomorphisms ofderivative graphs [38]. Locallysurjective homomorphisms are alsocalled

color dominations

[35].In additionthey areknownas

role assignments due

totheir applicationsinsocialscience[13,39,40].Justlikelocallybijective homomorphismstheyalsohaveapplicationsindistributedcomputing[9].

Ifthereexistsahomomorphismfromagraph

G to

agraph

H that

islocallybijective,locallyinjective,orlocallysurjective, respectively,thenwe write G

−→

B H , G

−→

I H , andG

−→

S H , respectively. Wedenotethedecisionproblemsthat aretotest whether

G

−→

B H , G

−→

I H , or

G

−→

S H for twogivengraphs G and H by LBHom, LIHom and LSHom, respectively.Allthree problemsare knowntobe NP-completewhen bothguestandhostgraphsaregivenasinput (seebelowfordetails),and attemptshavebeenmadetoclassifytheir computationalcomplexitywhenonly theguestgraphbelongstotheinputand thehostgraphisfixed.Thecorresponding problemsaredenotedby H -LBHom, H -LIHom, and H -LSHom, respectively.The

H -LSHom problem ispolynomial-timesolvableeitherifH has noedgeorif H is bipartiteandhasatleastoneconnected component isomorphic to an edge; in all other cases H -LSHom is NP-complete, even when the guest graph belongsto the class of bipartite graphs [20]. The complexity classification of H -LBHom and H -LIHom is still open, although many partial results are known forboth problems; we refer to the papers [1,6,16,18,33,34,36] andto the survey by Fiala and Kratochvíl[15]forboth NP-completeandpolynomiallysolvablecases.

Instead of fixing the host graph, another natural restriction is to only take guest graphs from a special graph class. Heggernesetal.[30]provedthat LBHom is GraphIsomorphism_{-completewhentheguestgraphischordal,and} polynomial-time solvablewhen theguestgraph isinterval. Incontrast, LSHom is NP-complete whenthe guestgraphis chordal and polynomial-timesolvablewhen theguest graphisproper interval, whereas LIHom is NP-completeeven forguestgraphs thatareproperinterval[30].Itisalsoknownthattheproblems LBHom and LSHom arepolynomial-timesolvablewhenthe guestgraphisatree[21].

Inthispaperwefocusonthefollowinglineofresearch.The

core of

agraph

G is

asubgraph F of G such that

G

→

F and

thereisnopropersubgraph Fof F with G

→

F.Itisknownthatthecoreofagraphisuniqueuptoisomorphism[28]. Dalmau,KolaitisandVardi[12]provedthatthe Hom problemispolynomial-timesolvablewhentheguestgraphbelongsto anyfixedclassofgraphswhosecoreshaveboundedtreewidth.Inparticular,thisresultimpliesan earlierresultthat Hom is polynomial-timesolvablewhen theguestgraph hasboundedtreewidth [11,22].Grohe [25] strengthened theresult of Dalmauetal.[12]by provingthat underacertain complexity assumption,namelyFPT

=

W

[

1

]

,the Hom problem canbe solvedinpolynomialtimeifandonlyifthisconditionholds.

ItisanaturalquestionwhethertheaboveresultsofDalmauetal.[12]andGrohe [25]remaintruewhenweconsider locallyconstrainedhomomorphismsinsteadofgeneralhomomorphisms.Wecanalreadyconcludefromknownresultsthat thisis not the casefor locally surjective homomorphisms. Recallthat H -LSHom is NP-complete even for bipartite guest graphsif H contains atleastoneedgeandiseithernon-bipartiteordoesnotcontain aconnectedcomponentisomorphic to an edge[20].The coreofevery bipartite graph withatleastone edge isan edge, andconsequently, hastreewidth 1. Thismeansthatbipartitegraphsformaclassofgraphswhosecoreshaveboundedtreewidth.Duetothisnegativeanswer, we turnourattentionto classesofgraphs ofboundedtreewidth insteadofclasses ofgraphswhose coreshavebounded treewidth,andposethefollowing(weaker)questioninstead:

Are LBHom, LIHom and LSHom polynomial-time solvable when the guest graph belongs to a class of bounded treewidth?

Thisquestionisfurthermotivatedbytwoknownresults,namelythat LBHom and LSHom canbothbesolvedinpolynomial timeiftheguestgraphisatree,andconsequentlyongraphsoftreewidth1[21].

1.1. Our contribution

InSection 3,we providea negativeanswer tothisquestion by showingthatthe problems LBHom, LSHom and LIHom are NP-complete alreadyinthe restrictedcasewherethe guestgraphhaspathwidthatmost 5,4 or2, respectively.We

also show that the three problems are NP-complete even if both the guest graph and the host graph have maximum

degree 3.Thelatterresultshowsthatlocallyconstrainedhomomorphismsproblemsbehavemorelikeunconstrained homo-morphismsongraphs ofboundeddegreethanongraphsofboundedtreewidth,asitisknownthat,forexample,

C5

-Hom is NP-completeonsubcubicgraphs[23].

Onthepositiveside, inSection 4,weshow thatall threeproblemscanbesolved inpolynomialtime ifwe boundthe treewidth of the guest graphand at the sametime bound the maximum degree of the guestgraph or the hostgraph. Becauseagraphclassofboundedmaximumdegreehasboundedtreewidthifandonlyifithasboundedclique-width[26], all three problemsare alsopolynomial-time solvablewhen we bound theclique-widthand themaximum degree ofthe

guestgraph.InSection4wealsoshowthat LIHom canbesolvedinpolynomialtimewhentheguestgraphhastreewidth 1, whichisbestpossiblegiventhehardnessresultfor LIHom showninSection3.

InSection5westatesomerelevantopenproblems. 2. Preliminaries

Let

G be

agraph.The

degree of

avertexv in G is denotedby

d

G

(

v

)

= |

NG

(

v

)

|

,and

(

G

)

=

maxv∈VGdG

(

v

)

denotesthe maximumdegreeof

G.

Let

ϕ

beahomomorphismfrom

G to

agraph

H .

Moreover,let

G

beaninducedsubgraphof

G,

and let

ϕ

 beahomomorphismfrom

G

to H . Wesaythat

ϕ

extends (or, equivalently,isan

extension of) ϕ

if

ϕ

(

v

)

=

ϕ



(

v

)

for every

v

∈

VG.

A

tree decomposition of G is

atree

T

= (

VT

,

E

T

)

,wheretheelementsof

V

T,calledthe

nodes of T ,

aresubsetsof

V

Gsuch

thatthefollowingthreeconditionsaresatisfied:

1. foreachvertex

v

∈

VG,thereisanode X

∈

VT withv

∈

X ,

2. foreachedge

uv

∈

EG,thereisanode X

∈

VT with

{

u

,

v

}

⊆

X ,

3. foreachvertex

v

∈

VG,thesetofnodes

{

X

|

v

∈

X

}

inducesaconnectedsubtreeof

T .

The

width of

atreedecomposition T is thesize ofalargestnode X minus one. The

treewidth of G,

denotedby tw

(

G

)

,is theminimumwidthoverallpossibletreedecompositionsof G.A path

decomposition of G is

atreedecomposition T of G

where

T is

apath.The

pathwidth of G is

theminimumwidthoverallpossiblepathdecompositionsof G.Bydefinition,the pathwidthof

G is

atleastashighasitstreewidth.Atreedecomposition

T is nice

[31]if

T is

abinarytree,rootedinaroot

R such thatthenodesof

T belong

tooneofthefollowingfourtypes: 1. a

leaf node X is

aleafof

T ,

2. an

introduce node X has

onechild

Y and

X

=

Y

∪ {

v

}

forsomevertex

v

∈

VG

\

Y ,

3. a

forget node X has

onechild

Y and

X

=

Y

\ {

v

}

forsomevertex

v

∈

Y ,

4. a

join node X has

twochildren

Y

,

Z satisfying

X

=

Y

=

Z .

An

equitable partition of

aconnectedgraph

G is

apartitionofitsvertexsetinblocks B1

,

. . . ,

Bksuchthatanyvertexin Bi hasthesame numbermi,j ofneighborsin Bj.Wecall thematrix M

= (

mi,j

)

corresponding to thecoarsest equitable

partitionof

G (in

whichtheblocksareorderedinsomecanonicalway;cf.[2])the

degree refinement matrix of G,

denotedas drm

(

G

)

.InSection3,wewillmakeuseofthefollowinglemma;aproofofthefirststatementinthislemmacanbefound inthepaperofFialaandKratochvíl[16],whereasthesecondstatementisduetoKristiansenandTelle[35].

Lemma1.

Let G and H be two graphs. Then the following two statements hold:

(i) if

G

−

→

I H and drm

(

G

)

=

drm

(

H

)

, then G

−

→

B H ;

(ii) if

G

₋

_→

S _{H and}

drm

(

G

)

=

drm

(

H

)

, then G

₋

_→

B H .

3. NP-completenessresults

Forthe NP-hardnessresultsin

Theorem 1

belowweuseareductionfromthe3-Partition problem.Thisproblemtakesas inputamultiset A of 3m integers,denotedinthesequelby

{

a1

,

a2

,

. . . ,

a3m

}

,andapositiveinteger

b,

suchthat b₄

<

ai

<

b₂

for all i

∈ {

1

,

. . . ,

3m

}

and



_1≤i≤3mai

=

mb. The taskisto determine whether A can be partitionedinto m disjoint sets A1

,

. . . ,

A

m suchthat



a∈Aia

=

b for all

i

∈ {

1

,

. . . ,

m

}

.Notethat therestrictions onthesizeofeachelementin A implies that eachset Ai inthedesiredpartitionmustcontainexactlythreeelements,whichiswhysuch apartition A1

,

. . . ,

A

m is

called a3-partition of A. The3-Partition problem isstrongly NP-complete [24], i.e.,it remains NP-complete even ifthe problemisencodedinunary.

Theorem1.

The following three statements hold:

(i) LBHom

is

NP-complete on input pairs

(

G

,

H

)

where G has pathwidth at most 5 and H has pathwidth at most 3;

(ii) LSHom

is

NP-complete on input pairs

(

G

,

H

)

where G has pathwidth at most 4 and H has pathwidth at most 3;

(iii) LIHom

is

NP-complete on input pairs

(

G

,

H

)

where G has pathwidth at most 2 and

H has pathwidth at most 2.

Proof. Firstnotethatallthreeproblemsarein NP.Weproveeachstatementseparatelyandstartwithstatement(i). Givenaninstance

(

A

,

b

)

of3-Partition,weconstructtwographs

G and H as

follows;see

Figs. 1 and 2

forsomehelpful illustrations.Theconstruction ofG starts bytaking3m disjointcyclesC1

,

. . . ,

C3m oflength

b,

one foreachelement of A.

Foreach

i

∈ {

1

,

. . . ,

3m

}

,theverticesof

C

i arelabeled

u

i1

,

. . . ,

uib andwe add,foreach j

∈ {

1

,

. . . ,

b

}

,two newvertices pij

and

q

i

j aswellastwonewedges

u

i jp i j and

u

i jq i

Fig. 1. A

schematic illustration of the graphs

G andH that

are constructed from a given instance

(A,b)of 3-Partition in the proof of statement (i) in

Theorem 1. See also Fig. 2for a more detailed illustration of the “leftmost” part of G and

the “rightmost” part of

H ,

including more labels.

Fig. 2. More detailed illustration of parts of the graphs G and H inFig. 1.

vertices

p

i

1

,

p

i2

,

. . . ,

piai and

q

i

1

,

qi2

,

. . . ,

qiai forevery

i

∈ {

1

,

. . . ,

3m

}

.Finally,thevertex y is madeadjacenttoeveryvertexp i

j

that isnotadjacentto x, andthevertex z is madeadjacentto everyvertex

q

i

j that isnotadjacentto x. Thisfinishesthe

constructionof G.

To construct H , we take m disjoint cycles C1

˜

,

. . . ,

C

˜

m of length b, where the vertices of each cycle C

˜

i are labeled

˜

ui₁

,

. . . ,

u

˜

i_b. For each i

∈ {

1

,

. . . ,

m

}

and j

∈ {

1

,

. . . ,

b

}

, we add two vertices p

˜

i_j and q

˜

i_j and make both of them adjacent tou

˜

i

j.Finally,weaddavertexx and

˜

makeitadjacenttoeachoftheverticesp

˜

ijandq

˜

ij.Thisfinishestheconstructionof H .

Wenowshowthatthereexistsalocallybijectivehomomorphismfrom

G to

H if andonlyif

(

A

,

b

)

isayes-instanceof 3-Partition.

Letusfirstassume thatthereexists alocallybijective homomorphism

ϕ

fromG to H . Since

ϕ

isa degree-preserving mapping,wemusthave

ϕ

(

x

)

= ˜

x. Moreover,since

ϕ

islocallybijective,therestrictionof

ϕ

to

N

G

(

x

)

isabijectionfrom

N

G

(

x

)

to NH

(

x

˜

)

.Againusingthedefinitionofalocallybijectivemapping,thistimeconsideringtheneighborhoodsofthevertices

in

N

H

(

˜

x

)

,wededucethatthereisabijectionfromtheset

N

G2

(

x

)

:= {

uij

|

1

≤

i

≤

3m

,

1

≤

j

≤

ai

}

,i.e.,fromthesetofvertices

in

G at

distance2 from

x,

totheset N2

H

(

˜

x

)

:= {˜

ukj

|

1

≤

k

≤

m

,

1

≤

j

≤

b

}

ofverticesthatareatdistance2 fromx in H .

˜

For

every

k

∈ {

1

,

. . . ,

m

}

,wedefineaset Ak

⊆

A such that Akcontainselement

a

i

∈

A if andonlyif

ϕ

(

ui1

)

∈ {˜

uk1

,

. . . ,

u

˜

kb

}

.Since

ϕ

isabijectionfrom

N

2

G

(

x

)

to

N

2H

(

x

˜

)

,thesets

A1

,

. . . ,

A

maredisjoint;moreovereachelement

a

i

∈

A is containedinexactly

oneofthem.Observethatthesubgraph of

G induced

by N2

G

(

x

)

isadisjointunionof3m pathsoflengths

a1

,

a2

,

. . . ,

a3m,

respectively,whilethesubgraphof

H induced

by N2_H

(

˜

x

)

isadisjointunionof

m cycles

oflength

b each.

Thefactthat

ϕ

is ahomomorphismandthereforenevermapsadjacentverticesof

G to

non-adjacentverticesin H implies that



_a∈A

ia

=

b forall

i

∈ {

1

,

. . . ,

m

}

.Hence A1

,

. . . ,

A

m isa3-partitionof

A.

Forthereversedirection,supposethereexistsa3-partition A1

,

. . . ,

A

mofA. Wedefineamapping

ϕ

asfollows.Wefirst

set

ϕ

(

x

)

=

ϕ

(

y

)

=

ϕ

(

z

)

= ˜

x. Let Ai

= {

ar

,

as

,

at

}

beanysetofthe 3-partition.We mapthe verticesofthecycles Cr

,

Cs

,

Ct

that are at distance 2 from x to the vertices of the cycle C

˜

i in the following way:

ϕ

(

urj

)

= ˜

u i

j foreach j

∈ {

1

,

. . . ,

ar

}

,

ϕ

(

us_j

)

= ˜

ui_a

r+j foreach j

∈ {

1

,

. . . ,

as

}

,and

ϕ

(

u t

j

)

= ˜

u i

ar+as+j foreach j

∈ {

1

,

. . . ,

at

}

. Theverticesof

C

r,

C

s and

C

t thatare atdistancemore than2 from x in G are mappedto verticesof C

˜

i suchthat the verticesof Cr, Cs andCt appearinthe

and Ct thatare atdistancemorethan 2 from x are mappedto verticesofC

˜

i analogously.Afterthe verticesofthecycles C1

,

. . . ,

C3mhavebeenmappedinthewaydescribedabove,itremainstomapthevertices

p

ijand

q

ijforeach

i

∈ {

1

,

. . . ,

3m

}

and j

∈ {

1

,

. . . ,

b

}

. Let pi

j

,

qij bea pairofvertices inG that are adjacentto x, andlet uij bethe second commonneighborof pij andqij.

Suppose u

˜

k_ istheimage of

u

i

j,i.e.,supposethat

ϕ

(

uij

)

= ˜

uk.Then wemap pij and

q

ij to p

˜

k andq

˜

k,respectively. Wenow consider the neighborsof y and z in G. Byconstruction, the neighborhood of y consists ofthe 2mb vertices inthe set

{

pi_j

|

ai+1

≤

j

≤

b

}

,whileNG

(

z

)

= {

qi_j

|

ai+1

≤

j

≤

b

}

.

Observethatx,

˜

theimageof y and z, isadjacenttotwosetsof

mb vertices:

oneoftheformp

˜

k_,theotheroftheformq

˜

k_. Hence, we needtomap halftheneighborsof y to verticesofthe formp

˜

k_ andhalf theneighborsof y to verticesofthe formq

˜

k_inordertomake

ϕ

alocallybijectivehomomorphism.Thesameshouldbedonewiththeneighborsof

z.

Forevery vertexu

˜

k_in H , wedoasfollows.Byconstruction,exactlythreeverticesofG are mappedtou

˜

k_,andexactlytwoofthose vertices, say ui_j and u_hg, are atdistance 2 from y in G. We set

ϕ

(

pi_j

)

= ˜

p_k and

ϕ

(

p_hg

)

= ˜

qk_.We also set

ϕ

(

qi_j

)

= ˜

qk_ and

ϕ

(

qg_h

)

= ˜

pk_.Thiscompletesthedefinitionofthemapping

ϕ

.

Since themapping

ϕ

preservesadjacencies, itclearly isahomomorphism.In ordertoshow that

ϕ

islocallybijective, wefirstobservethatthedegreeofeveryvertexin

G is

equaltothedegreeofitsimageinH ; inparticular,

d

G

(

x

)

=

dG

(

y

)

=

dG

(

z

)

=

dH

(

x

˜

)

=

2mb.Fromtheabovedescriptionof

ϕ

wegetabijectionbetweentheverticesof

N

H

(

˜

x

)

andtheverticesof NG

(

v

)

foreach

v

∈ {

x

,

y

,

z

}

.Foreveryvertex

p

i_jthatisadjacentto

x and u

i_jin

G,

itsimage p

˜

k_isadjacenttotheimagesx of

˜

x and u

˜

k_of

u

i

j.Foreveryvertex

p

i

j thatisadjacentto y (respectively z) and

u

i jin

G,

itsimage p

˜

k  orq

˜

k  isadjacenttox of

˜

y (respectively z) andu

˜

k

of

u

ij.Hencetherestrictionof

ϕ

to

N

G

(

pij

)

isbijectiveforevery

i

∈ {

1

,

. . . ,

3m

}

and j

∈ {

1

,

. . . ,

b

}

,

andthesameclearly holdsfortherestrictionof

ϕ

to NG

(

qi_j

)

.Theverticesofeachcycle

C

i are mappedtotheverticesof

somecycleC

˜

k insuchawaythattheverticesandtheirimagesappearinthesameorderonthecycles.This,togetherwith

thefactthattheimageu

˜

k_ofeveryvertex

u

i

jisadjacenttotheimages p

˜

k andq

˜

k  oftheneighbors

p

i j and

q

i j of

u

i j,shows

that therestrictionof

ϕ

to NG

(

ui_j

)

isbijectiveforevery

i

∈ {

1

,

. . . ,

3m

}

and j

∈ {

1

,

. . . ,

b

}

.Weconcludethat

ϕ

isalocally

bijectivehomomorphismfrom

G to H .

Inordertoshowthatthepathwidthof

G is

atmost 5,letusfirstconsiderthesubgraphof

G depicted

ontheleft-hand side of

Fig. 2

; wedenotethissubgraph by L1,andwesaythatthecycle

C1

defines the subgraph L1.The graphL₁ thatis obtainedfrom

L1

bydeletingvertices

x

,

y

,

z and

edge

u

1

1u1bisacaterpillar,i.e.,atreeinwhichthereisapathcontainingall

verticesofdegreemorethan1.Sincecaterpillarsarewell-knowntohavepathwidth1,graphL₁ hasapathdecomposition

P₁ ofwidth 1.Startingwith P₁,we cannowobtainapathdecompositionofthegraphL1 bysimplyaddingvertices

x,

y, z and u1

1 toeachnodeof P1;thispathdecompositionhaswidth 5.EverycycleCi in

G defines

asubgraph Li of

G in

the

sameway

C

1 definesthesubgraph

L

1.Supposewehaveconstructedapathdecomposition Pi ofwidth 5 ofthesubgraph Li foreach

i

∈ {

1

,

. . . ,

3m

}

inthewaydescribedabove.Sinceanytwosubgraphs

L

iand

L

jwith

i

=

j have onlythevertices x

,

y

,

z in common, andthesethreeverticesappearinallnodesofeachofthepathdecompositions Pi,wecanarrangethe

3m pathdecompositions P1

,

. . . ,

P

3m insuchawaythatweobtainapathdecomposition P of G of width 5.Hence

G has

pathwidthatmost 5.Similarbuteasierargumentscanbeusedtoshowthat H has pathwidthatmost 3.

The NP-hardness reductionforthelocallybijectivecasecanalsobe usedtoprovethat LIHom and LSHom are NP-hard forinputpairs

(

G

,

H

)

whereG has pathwidthatmost 5 and

H has

pathwidthatmost 3.Thisfollowsfromtheclaimthat

G

−→

B H if andonlyif

G

−→

S H if andonlyif

G

−→

I H for thegadgetgraphs

G and

H displayed in

Fig. 1

.Thisclaimcanbe seenasfollows.Firstsupposethat

G

−→

B H . Then,bydefinition,

G

−→

S H and G

−→

I H . Nowsupposethat

G

−→

I H or G

−→

S H .

Sinceitcaneasilybeverifiedthat

drm

(G

)

=

drm

(H

)

=



_{0 0 2mb} 0 2 2 1 1 0



,

wecanuse

Lemma 1

(i)or(ii),respectively,todeducethat

G

−→

B H . However,wecanstrengthenthehardnessresultsforthe locallysurjectiveandinjectivecasesbyreducingthepathwidthoftheguestgraphtobeatmost4 and 2,respectively,and inthelattercasewecansimultaneouslyreducethepathwidthofthehostgraphtobeatmost 2,asclaimedinstatements (ii)and(iii)of

Theorem 1

.Inordertodoso,wegivethefollowingalternativeconstructionsbelow.

The alternative hardness construction for LSHom is similar to buteasierthan the construction for LBHom;see Fig. 3. Let

(

A

,

b

)

beaninstanceof3-Partition.Weconstructagraph

G

bytaking3m disjointcycles

C1

,

. . . ,

C3m oflength b,and labelingtheverticesofeachcycle

C

iwithlabels

u

i1

,

. . . ,

uiBinthesamewayaswelabeledtheverticesofthecycles

C

iinthe

construction for LBHom (seealso

Fig. 2

).Wethenaddtwovertices

x and

y. Forevery

i

∈ {

1

,

. . . ,

3m

}

,wemake

x adjacent

to eachofthe verticesui₁

,

ui₂

,

. . . ,

u_ai_i,and y is madeadjacenttoeach ofthevertices

u

i

ai+1

,

. . . ,

u i

B.Graph H isobtained

fromthedisjointunionof

m cycles

C1

˜

,

. . . ,

C

˜

moflength

b by

addingoneuniversalvertexx.

˜

Usingsimilarargumentsasthe

onesusedinthe NP-hardnessproofof LBHom,itcanbeshownthatthereexistsalocallysurjectivehomomorphism

ϕ

from

Gto H ifandonlyif

(

A

,

b

)

isa yes-instanceof3-Partition.Suchahomomorphism

ϕ

 maps

x and

y to x,

˜

asotherwise a neighbor of

x or

y, whichhasdegree 3, mustbe mappedto x which

˜

hasdegreelarger than 3.Moreover,

ϕ

 mapsthe

Fig. 3. A

schematic illustration of the graphs

Gand Hthat are constructed from a given instance (A,b)of 3-Partition in the proof of statement (ii) in

Theorem 1.

Fig. 4. A

schematic illustration of the graphs

Gand Hthat are constructed from a given instance (A,b)of 3-Partition in the proof of statement (iii) in

Theorem 1.

verticesofcycles

C

1

,

. . . ,

C3mtotheverticesofcyclesC

˜

1

,

. . . ,

C

˜

minexactlythesamewayas

ϕ

mappedtheseverticesinthe

NP-hardnessproofof LBHom.Itisa routineexercisetoshowthat Ghaspathwidthatmost 4 andthat Hhaspathwidth atmost 3.

The reduction for LIHom iseven easier;see Fig. 4. Givenan instance

(

A

,

b

)

of3-Partition, we createa graph G by addingauniversalvertex

x to

thedisjointunionof3m pathson

a1

,

a2

,

. . . ,

a3mvertices,respectively.Graph H isobtained fromthedisjointunionof

m paths

on

b vertices

byaddingauniversalvertex

˜

x. Itiseasytoverifythatthereexistsalocally injectivehomomorphism

ϕ

 from

G

 to H,mapping

x to

x and

˜

allotherverticesof

G

 totheverticesofdegree 2 or3 in

H,ifandonlyif

(

A

,

b

)

isayes-instanceof3-Partition.Theobservationthatboth

G

 andH havepathwidth 2 completes theproofof

Theorem 1

.

2

Wenowconsiderthecasewhereweboundthemaximumdegreeof

G instead

ofthetreewidthofG. Wewillcombine some knownresultsinorderto showthat boundingthemaximumdegreeof G does not yieldtractability foranyofour

threeproblems LBHom, LIHom and LSHom.

Kratochvíl andKˇrivánek [32] showed that K4-LBHom _{is NP-complete,where K4 denotes the complete graph onfour} vertices.Sinceagraph

G allows

alocallybijectivehomomorphismto

K

4onlyif

G is

3-regular,

K

4-LBHom is NP-completeon

3-regulargraphs.Thedegreerefinementmatrixofa3-regulargraphisthe1

×

1 matrixwhoseonlyentryis 3.Consequently, dueto Lemma 1, K4-LBHom is equivalent to K4-LIHom andto K4-LSHom on 3-regular graphs. Thisyields thefollowing result.

Theorem2.

The problems LBHom, LIHom and LSHom are

NP-complete on input pairs

(

G

,

K

4

)

where G has maximum degree 3. Theorem 2is tightinthefollowing sense.Allthree problems LBHom, LIHom and LSHom arepolynomial-timesolvable oninputpairs

(

G

,

H

)

where

G has

maximumdegreeatmost 2.Moreover,thefirsttwoproblemsarealsopolynomial-time solvableoninputgraphs

(

G

,

H

)

whereonly

H has

maximumdegreeatmost 2,as

G is

requiredtobeofmaximumdegree atmost 2 aswellinthesetwocases.Thisdoesnotholdfor LSHom,as

K3

-LSHom is NP-complete[35].

4. Polynomial-timeresults

InSection3,weshowedthat LBHom, LIHom and LSHom are NP-completewheneitherthetreewidthorthemaximum degree oftheguestgraphisbounded.Inthissection,we showthat all threeproblemsbecomepolynomial-timesolvable ifweboundboththetreewidthandthemaximumdegreeof G.Fortheproblems LBHom and LIHom,ourpolynomial-time resultfollows fromreformulating these problemsasconstraint satisfactionproblems andapplyinga result ofDalmauet al.[12].Inordertoexplainthis,weneedsomeadditionalterminology.

Arelationalstructure

(

A

,

R1

,

. . . ,

R

k

)

isafiniteset A, calledthe

base

set, togetherwithacollectionofrelations

R1

,

. . . ,

R

k.

Thearitiesoftheserelationsdeterminethevocabularyofthestructure.Ahomomorphismbetweentworelationalstructures ofthesamevocabularyisamappingbetweenthebasesetssuchthatalltherelationsarepreserved.

Fiala andKratochvíl [14] observedthat locallyinjective andlocally bijective homomorphisms betweengraphs can be expressedashomomorphismsbetweenrelationalstructuresasfollows.Alocallyinjectivehomomorphism f

:

G

→

H can be expressedasahomomorphismbetweenrelationalstructures

(

VG

,

E

G

,

EG

)

and

(

VH

,

EH

,

EH

)

,wherethenewbinaryrelation Econsistsofpairsofdistinctverticesthathaveatleastonecommonneighbor.Since f maps distinctneighborsofavertex

v to distinct neighbors of f

(

v

)

,we getthat f is a homomorphismof the associatedrelational structures. Onthe other hand, if

(

VG

,

E

G

,

E

_G

)

and

(

VH

,

E

H

,

E

_H

)

are constructed fromG and H as described above,andif f is a homomorphism

betweenthem,thentherelations E_G andE_H guaranteethatnotwoverticeswithacommonneighborin

G are

mappedto the sametarget in H .Inother words, f is alocallyinjectivehomomorphismbetweenthegraphs G and H . Ananalogous constructionworksforlocallybijectivehomomorphisms.Here,weneedtoexpress

G using

twobinaryrelations

E

G andEG

asabove,togetherwith

(

G

)

+

1 unaryrelations.Aunaryrelationcanbeviewedasaset:here,the

i-th

setwillconsistsof allverticesofdegree i

−

1.Theseunaryrelationsguaranteethatdegreesarepreserved,andconsequentlythattheassociated graphhomomorphismsarelocallybijective.

The

Gaifman graph

G

Aofarelationalstructure

A

= (

A

,

R1

,

. . . ,

Rk

)

isthegraphwithvertexset

A,

whereanytwodistinct

vertices u and v are joinedby an edgeif theyare bound by some relation.Formally u

,

v

∈

E_GA if andonlyifforsome relation Riofarity

r and

(

a1

,

. . . ,

ar

)

∈

Riitholdsthat

{

u

,

v

}

⊆ {

a1

, . . . ,

ar

}

.

As a directconsequence of a resultof Dalmauet al. [12], the existence ofa homomorphismbetween two relational structures

A

and

B

can be decided in polynomial time if the treewidth of

G

_A is bounded by a constant. Thisleads to

Theorem 3below.

Theorem3.

The problems LBHom and LIHom can be solved in polynomial time when G has bounded treewidth and G or

H has bounded maximum degree.

Proof. First supposethat G has boundedtreewidth andboundedmaximumdegree. Observethatforlocallyinjectiveand locallybijectivehomomorphisms,theGaifmangraph

G

_Aisisomorphicto

G

2_{,whichisthegrapharisingfrom}

_{G by}

_adding

an edge betweenany two vertices at distance 2. It suffices to observe that tw

(

G2

₎

_{≤ (}

_G

₎₍

_tw

₍

_G

₎

₊

₁

₎

₋

_{1, as we can}

transformanytreedecomposition

T of G of

widthtw

(

G

)

intoadesiredtreedecompositionofG2 _{byaddingtoeachnode} X of T all theneighborsofevery vertexfrom X . Since G

−→

I H implies that

(

H

)

≥ (

G

)

,the theoremalsoholds ifwe

boundthemaximumdegreeofH instead of G.

2

To our knowledge,locally surjective homomorphisms have not yet beenexpressed ashomomorphisms between rela-tionalstructures.Hence,intheproofof

Theorem 4

below,wepresentapolynomial-timealgorithmfor LSHom when G has

boundedtreewidthandboundedmaximumdegree.Wefirstintroducesomeadditionalterminology.

Let

ϕ

bealocallysurjectivehomomorphismfrom

G to

H . Letv

∈

VG andp

∈

VH.If

ϕ

(

v

)

=

p, i.e.,if

ϕ

mapsvertex v

tocolor p, thenwesaythat

p is assigned to

v. Bydefinition,foreveryvertex

v

∈

VG,thesetofcolorsthatareassignedto

theneighborsof

v in G is

exactlytheneighborhoodof

ϕ

(

v

)

in H . Nowsupposewearegivenahomomorphism

ϕ

froman inducedsubgraph

G

of

G to H .

Foranyvertex

v

∈

VG,wesaythat

v misses a

colorp

∈

VH if p

∈

NH

(

ϕ



(

v

))

\

ϕ



(

NG

(

v

))

,

i.e.,if

ϕ

doesnotassign

p to

anyneighborof

v in G

,butanylocallysurjectivehomomorphism

ϕ

from

G to H that

extends

ϕ

 assigns

p to

someneighborofv in G.

Let T be a nicetreedecomposition ofG rooted in R. Foreverynode X

∈

VT,we define GX to bethe subgraphof G

induced bytheverticesof X together withthe verticesofall thenodesthatare descendantsof X . Inparticular, wehave

GR

=

G.

Definition1.Let X

∈

VT,andlet

c

:

X

→

VH and

μ

:

X

→

2VH betwo mappings.The pair

(

c

,

μ

)

is

feasible for G

X ifthere

existsahomomorphism

ϕ

from

G

X to H satisfying thefollowingthreeconditions:

(i) c

(

v

)

=

ϕ

(

v

)

forevery v

∈

X ;

(ii)

μ

(

v

)

=

NH

(

ϕ

(

v

))

\

ϕ

(

NGX

(

v

))

forevery v

∈

X ; (iii)

ϕ

(

NG

(

v

))

=

NH

(

ϕ

(

v

))

forevery

v

∈

VGX

\

X .

Inotherwords,apair

(

c

,

μ

)

consistsofacoloring

c of

theverticesof X , togetherwithacollectionofsets

μ

(

v

)

,onefor each

v

∈

X , consistingofexactlythosecolorsthat v misses. Informallyspeaking,apair

(

c

,

μ

)

isfeasibleforGX ifthereis

a homomorphism

ϕ

:

GX

→

H such that

ϕ

“agrees”withthecoloring c on theset X , andsuch that noneofthevertices

in VGX

\

X misses anycolor. The idea is that ifa pair

(

c

,

μ

)

is feasible, then such a homomorphism

ϕ

might havean extension

ϕ

∗ that isa locallysurjective homomorphism fromG to H . Afterall, foranyvertex v

∈

X that missesa color whenconsidering

ϕ

,thiscolormightbeassignedby

ϕ

∗toaneighborofv in theset

V

G

\

VGX.

Theorem4.

The

problem LSHom can be solved in polynomial time when G has bounded treewidth and G or H has bounded maximum degree.

Proof. Let

(

G

,

H

)

beaninstanceof LSHom suchthatthetreewidthoftheguestgraph

G is

bounded.Throughouttheproof,

we assume that the maximum degree of H is bounded, and show that the problemcan be solved in polynomial time

undertheserestrictions.Since

G

_−→

S _{H implies}

that

(

G

)

≥ (

H

)

,ourpolynomial-timeresultappliesalsoifweboundthe maximumdegreeof

G instead

of H .

We mayassume withoutloss ofgenerality that both G and H are connected,asotherwise we justconsiderall pairs

(

Gi

,

H

j

)

separately, where Gi is a connected component of G and Hj is a connected component of H . Because G has

bounded treewidth, we can compute a tree decomposition of G of width tw

(

G

)

inlinear time using Bodlaender’s algo-rithm [8]. We transformthistree decompositioninto anice treedecomposition T of G with width tw

(

G

)

withat most 4

|

VG

|

nodesusingthelinear-timealgorithmofKloks[31].Let

R be

therootof

T and

let

k

=

tw

(

G

)

+

1.

Foreachnode X

∈

VT,let FX bethesetofallfeasiblepairs

(

c

,

μ

)

for

G

X.Forevery feasiblepair

(

c

,

μ

)

∈

FX andevery v

∈

X , itholdsthat

μ

(

v

)

isasubsetof

N

H

(

c

(

v

))

.Since

|

X

|

≤

k and

|

NH

(

c

(

v

))

|

≤ (

H

)

k for every

v

∈

X and everymapping c

:

X

→

VH,thisimpliesthat

|

FX

|

≤ |

VH

|

k2(H)k foreach X

∈

VT.Asweassumedthatboth

k and

(

H

)

areboundedbya

constant,theset FX isofpolynomialsizewithrespectto

|

VH

|

.

The algorithmconsiders the nodesof T in a bottom-up manner,starting withthe leavesof T and processinga node

X

∈

VT only after its children have been processed. For every node X , the algorithm computes the set FX in the way

describedbelow.Wedistinguishbetweenfourdifferentcases.Thecorrectnessofeachofthecaseseasilyfollowsfromthe definitionofalocallysurjectivehomomorphismand

Definition 1

.

1. X isa leaf node of T . We considerallmappings

c

:

X

→

VH.Foreachmapping

c,

wecheckwhether

c is

ahomomorphism

from

G

X to H . Ifnot,thenwediscard

c,

asitcannot belongtoa feasiblepairduetocondition(i)in

Definition 1

.For

eachmapping

c that

isnotdiscarded,wecompute theuniquemapping

μ

satisfying

μ

(

v

)

=

NH

(

c

(

v

))

\

c

(

NGX

(

v

))

for each

v

∈

X , andweaddthepair

(

c

,

μ

)

to FX.Itfollowsfromcondition(ii)thattheobtainedset FX indeedcontainsall

feasiblepairsfor

G

X.Asthereisnovertexin

V

GX

\

X , everypair

(

c

,

μ

)

triviallysatisfiescondition(iii).Thecomputation of

F

X canbedonein O

(

|

VH

|

kk

((

H

)

+

k

))

timeinthiscase.

2. X is

a forget node.

Let

Y be

thechildofX in T , andlet

{

u

}

=

Y

\

X . Observethat

(

c

,

μ

)

∈

FX ifandonlyifthereexists

afeasiblepair

(

c

,

μ



)

∈

FY suchthat

c

(

v

)

=

c

(

v

)

and

μ

(

v

)

=

μ



(

v

)

forevery

v

∈

X , and

μ



(

u

)

= ∅

.Henceweexamine

each

(

c

,

μ



)

∈

FY andcheckwhether

μ



(

u

)

= ∅

issatisfied.Ifso,we firstrestrict

(

c

,

μ



)

on X to get

(

c

,

μ

)

andthen

weinserttheobtainedfeasiblepairinto FX.Thisprocedureneeds

O

(

|

FY

|

k

(

H

))

timeintotal.

3. X is

an introduce node.

Let Y be thechildof X in T , andlet

{

u

}

=

X

\

Y . Observethat

(

c

,

μ

)

∈

FX ifandonlyifthere

existsafeasiblepair

(

c

,

μ



)

∈

FY such that,forevery

v

∈

Y , itholdsthat

c

(

v

)

=

c

(

v

)

,

μ

(

v

)

=

μ



(

v

)

\

c

(

u

)

if

uv

∈

EG,

and

μ

(

v

)

=

μ



(

v

)

ifuv

∈

/

EG.Hence, foreach

(

c

,

μ



)

∈

FY, we consider all

|

VH

|

mappings c

:

X

→

VH that extend c. Foreach such extension

c,

we test whetherc is ahomomorphism from GX to H by checkingthe adjacencies of c

(

u

)

in H .Ifnot,thenwemaysafelydiscard

c due

tocondition(i)in

Definition 1

.Otherwise,wecomputetheunique mapping

μ

:

X

→

2VH _satisfying

μ

(v)

=



_N H

(c(u))

\

c(NGX

(u))

if v

=

u

μ



(v)

\

c(u) if v

=

u and uv

∈

EG

μ



(v)

if v

=

u and uv

∈

/

EG

,

andwe addthepair

(

c

,

μ

)

to FX;duetocondition(ii),thispair

(

c

,

μ

)

istheuniquefeasiblepaircontaining

c.

Com-putingtheset FX takesatmost O

(

|

FY

||

VH

|

k

(

H

))

timeintotal.

4. X is

a join node.

Let

Y and

Z be thetwochildrenof X in T . Observethat

(

c

,

μ

)

∈

FX ifandonlyifthereexistfeasible

pairs

(

c1

,

μ1

)

∈

FY and

(

c2

,

μ2

)

∈

FZ such that, for every v

∈

X , c

(

v

)

=

c1

(

v

)

=

c2

(

v

)

and

μ

(

v

)

=

μ

1

(

v

)

∩

μ

2

(

v

)

.

Hence the algorithm considers every combinationof

(

c1

,

μ1

)

∈

FY with

(

c2

,

μ2

)

∈

FZ andifthey agree on thefirst

component c,theothercomponent

μ

isdetermineduniquelybytakingtheintersectionof

μ1

(

v

)

and

μ2

(

v

)

forevery

v

∈

X . Thisprocedurecomputestheset FX in O

(

|

FY

||

FZ

|

k

(

H

))

timeintotal.

Finally,observe that a locallysurjective homomorphism from G to H exists ifandonly ifthere exists a feasiblepair

(

c

,

μ

)

for GR such that

μ

(

v

)

= ∅

for all v

∈

R. Since T has at most 4

|

VG

|

nodes, we obtain a total running time of O

(

|

VG

|(|

VH

|

k2(H)k

)

2k

(

H

))

.Asweassumedthatboth

k

=

tw

(

G

)

+

1 and

(

H

)

areboundedbyaconstant,ouralgorithm

runsinpolynomialtime.

2

Theproof of

Theorem 4

suggestan alternativeproof ofTheorem 3thatdoesnotrely onthegeneralresultby Dalmau etal.[12].Toseewhythisisthecase,firstobservethatwecansolve LIHom usingadynamicprogrammingapproachthat stronglyresemblestheonefor LSHom describedintheproofof

Theorem 4

;insteadofkeepingtrackofsets

μ

(

v

)

ofcolors thatavertex v

∈

X is missing,wekeeptrackofsets

α

(

v

)

ofcolorsthathavealreadybeenassignedtotheneighborsofa vertex v

∈

X . Thisisbecausein alocallyinjectivehomomorphism fromG to H , no colormaybeassignedto morethan oneneighborofanyvertex.Inthiswaywecanadjust

Definition 1

insuchawaythatitworksforlocallyinjectiveinstead oflocallysurjectivehomomorphisms.Tosolve LBHom inpolynomialtimewithoutusingtheresultbyDalmauetal.[12],it

sufficestoobservethat

(

G

,

H

)

isayes-instanceof LBHom ifandonlyifitisayes-instanceforboth LIHom and LSHom and that we cansolve thelattertwo problemsinpolynomial time usingdynamic programmingasexplained above.Weomit furtherdetails,butweexpectthatdynamicprogrammingalgorithmsofthiskindwillhavesmallerhiddenconstantsinthe runningtimeestimatesthanthemoregeneralmethodofDalmauetal.[12].

We concludethissectionwithonemorepolynomial-timeresult. Itisknownthattheproblems LBHom and LSHom are polynomial-timesolvablewhen G is atree[21],andconsequentlywhen

G has

treewidth 1.Weshowthatthesameholds forthe LIHom problem.

Theorem5.

The LIHom problem can be solved in polynomial time when G has treewidth 1, i.e., when G is a forest.

Proof. Let usfirststatesometerminologyandusefulknownresults.TheuniversalcoverTG ofaconnectedgraphG is the

unique tree(whichmayhavean infinitenumberofvertices)such thatthereisalocallybijectivehomomorphismfromTG

to

G.

Onewaytodefinethismappingisasfollows.Consider allfinitewalksin G that startfroman arbitraryfixedvertex in G and thatdonottraverse thesameedgeintwoconsecutivesteps.Eachsuch walkwillcorrespondtoa vertexof TG.

We lettwoverticesofTG beadjacentifandonlyifonecanbeobtainedfromtheother bydeletingthelast vertexofthe

walk. Thenthemapping fG thatmapseverywalktoitslast vertexisalocallybijectivehomomorphismfrom

T

G to G[2].

It isalso knownthat TG

=

G if andonlyif G is a tree[2].Moreover,foranytwo graphs G and H , G

−→

I H implies that TG

−→

I TH [19].

Now let

(

G

,

H

)

be aninstanceof LIHom where

G has

treewidth 1. Weassume,withoutlossofgenerality, thatboth G

and H are connected.In particular, G is atree.We claim that G

−→

I H if andonlyif TG

−→

I TH.The forwardimplication

followsfromabove.Toshowthebackward implication,suppose that TG

−→

I TH.Then G

−→

I TH,because

T

G

=

G. Let f be

a locallyinjectivehomomorphismfrom

G to

TH.Then,because

G

−→

I TH and TH

−→

B H , wehave G

−→

I H . Toexplainthis,

consider themapping f

:

VG

→

VH definedby f

(

u

)

=

fH

(

x

)

ifandonlyif f

(

u

)

=

x. Notice that f isa locallyinjective

homomorphism from G to H . The desiredresult follows from thisclaim combined with the fact that we can check in polynomialtimewhether

T

G

−→

I TH holdsfortwographs

G and H

[21].

2

5. Conclusion

Theorem 5statesthat LIHom canbesolvedinpolynomialtimewhentheguestgraphhastreewidth 1,while

Theorem 1

impliesthat theproblemis NP-completewhentheguestgraphhastreewidth 2.Thisshowsthatthebound onthe path-width in thethird statement ofTheorem 1 is best possible.We leave it as an open problemto determine whether the boundsonthepathwidthintheothertwostatementsof

Theorem 1

canbereducedfurther.

Weconcludethispaperwithsomeremarksontheparameterizedcomplexityoftheproblems LIHom, LSHom andLBHom. The hardness resultsinthispapershowthat allthree problemsarepara-NP-completewhen parameterizedbyeither the

treewidth of G or the maximum degree of G. Theorems 3 and 4 show that the problems are in XP when

parameter-ized jointly by thetreewidth of G and the maximumdegree of G. A naturalquestion is whetherthe problemsare FPT when parameterized by the treewidth of G and the maximum degree of G, i.e., whether they can be solved in time

f

(

tw

(

G

),

(

G

))

· (|

VG

|

+ |

VH

|)

O(1)forsomefunction f thatdoesnotdependonthesizesof

G and H .

Acknowledgements

We wouldliketothankIsoldeAdlerforposingtheresearch questionsthatwe addressedinourpaperandforhelpful discussions.ThefourthauthoralsothanksJanArneTelleforfruitfuldiscussions.

References

[1]J.Abello,M.R.Fellows,J.C.Stillwell,Onthecomplexityandcombinatoricsofcoveringfinitecomplexes,Australas.J.Combin.4(1991)103–112. [2]D.Angluin,Localandglobalpropertiesinnetworksofprocessors,in:Proc.STOC1980,1980,pp. 82–93.

[3]D.Angluin,A.Gardiner,Finitecommoncoveringsofpairsofregulargraphs,J.Combin.TheorySer.B30(1981)184–187. [4]N.Biggs,AlgebraicGraphTheory,CambridgeUniversityPress,1974.

[5]N.Biggs,Constructing5-arctransitivecubicgraphs,J.Lond.Math.Soc.(2)26(1982)193–200.

[6]O.Bílka,B.Lidický,M.Tesaˇr,Locallyinjectivehomomorphismtothesimpleweightgraphs,in:Proc.TAMC2011,in:LNCS,vol. 6648,2011,pp. 471–482. [7]H.L.Bodlaender,Theclassificationofcoveringsofprocessornetworks,J.ParallelDistrib.Comput.6(1989)166–182.

[8]H.L.Bodlaender,Alinear-timealgorithmforfindingtree-decompositionsofsmalltreewidth,SIAMJ.Comput.25 (6)(1996)1305–1317.

[9]J.Chalopin,Y.Métivier,W.Zielonka,Election,namingandcellularedgelocalcomputations,in:Proc.ICGT2004,in:LNCS,vol. 3256,2004,pp. 242–256. [10]S.Chaplick,J.Fiala,P.van’tHof,D.Paulusma,M.Tesar,Locallyconstrainedhomomorphismsongraphsofboundedtreewidthandboundeddegree,in:

Proc.FCT2013,in:LNCS,vol. 8070,2013,pp. 121–132.

[11]Ch.Chekuri,A.Rajaraman,Conjunctivequerycontainmentrevisited,in:Proc.5thInternationalConferenceonDatabaseTheory,in:LNCS,vol. 1186, 1997,pp. 56–70.

[12]V.Dalmau,P.G.Kolaitis,M.Y.Vardi,Constraintsatisfaction,boundedtreewidth,andfinite-variablelogics,in:Proc.CP2002,in:LNCS,vol. 2470,2002, pp. 223–254.

[13]M.G.Everett,S.Borgatti,Rolecoloringagraph,Math.SocialSci.21(1991)183–188.

[14]J.Fiala,J.Kratochvíl,Locallyinjectivegraphhomomorphism:listsguaranteedichotomy,in:Proc.WG2006,in:LNCS,vol. 4271,2006,pp. 15–26. [15]J.Fiala,J.Kratochvíl,Locallyconstrainedgraphhomomorphisms–structure,complexity,andapplications,Comput.Sci.Rev.2(2008)97–111.

[16]J.Fiala,J.Kratochvíl,Partialcoversofgraphs,Discuss.Math.GraphTheory22(2002)89–99.

[17]J.Fiala,J.Kratochvíl,T.Kloks,Fixed-parametercomplexityofλ-labelings,DiscreteAppl.Math.113(2001)59–72.

[18]J.Fiala,J.Kratochvíl,A.Pór,Onthecomputationalcomplexityofpartialcoversofthetagraphs,DiscreteAppl.Math.156(2008)1143–1149. [19]J.Fiala,J.Maxová,Cantor–Bernsteintypetheoremforlocallyconstrainedgraphhomomorphisms,EuropeanJ.Combin.27(2006)1111–1116. [20]J.Fiala,D.Paulusma,Acompletecomplexityclassificationoftheroleassignmentproblem,Theoret.Comput.Sci.349(2005)67–81. [21]J.Fiala,D.Paulusma,Comparinguniversalcoversinpolynomialtime,TheoryComput.Syst.46(2010)620–635.

[22]E.C.Freuder,Complexityofk-treestructuredconstraintsatisfactionproblems,in:Proc.8thNationalConferenceonArtificialIntelligence,1990,pp. 4–9. [23]A.Galluccio,P.Hell,J.Nešetˇril,ThecomplexityofH -colouringofboundeddegreegraphs,DiscreteMath.222(2000)101–109.

[24]M.R.Garey,D.S.Johnson,ComputersandIntractability:AGuidetotheTheoryof NP-completeness,W.H.Freeman&Co.,NewYork,1979. [25]M.Grohe,Thecomplexityofhomomorphismandconstraintsatisfactionproblemsseenfromtheotherside,J.ACM54 (1)(2007).

[26]F.Gurski,E.Wanke,Thetree-widthofclique-widthboundedgraphswithoutKn,n,in:Proc.WG2000,in:LNCS,vol. 1928,2000,pp. 196–205. [27]P.Hell,J.Nešetˇril,OnthecomplexityofH -colouring,J.Combin.TheorySer.B48(1990)92–110.

[28]P.Hell,J.Nešetˇril,Thecoreofagraph,DiscreteMath.109(1992)117–126. [29]P.Hell,J.Nešetˇril,GraphsandHomomorphisms,OxfordUniversityPress,2004.

[30]P.Heggernes,P.van’tHof,D.Paulusma,Computingroleassignmentsofproperintervalgraphsinpolynomialtime,J.DiscreteAlgorithms14(2012) 173–188.

[31]T.Kloks,Treewidth,ComputationsandApproximations,LNCS,vol. 842,Springer,1994.

[32]J.Kratochvíl,M.Kˇrivánek,Onthecomputationalcomplexityofcodesingraphs,in:Proc.MFCS1988,in:LNCS,vol. 324,1988,pp. 396–404. [33]J.Kratochvíl,A.Proskurowski,J.A.Telle,Coveringregulargraphs,J.Combin.TheorySer.B71(1997)1–16.

[34]J.Kratochvíl,A.Proskurowski,J.A.Telle,Complexityofgraphcoveringproblems,NordicJ.Comput.5(1998)173–195. [35]P.Kristiansen,J.A.Telle,GeneralizedH -coloringofgraphs,in:Proc.ISAAC2000,in:LNCS,vol. 1969,2000,pp. 456–466.

[36]B.Lidický,M.Tesaˇr,Complexityoflocallyinjectivehomomorphismtothethetagraphs,in:Prov.IWOCA2010,in:LNCS,vol. 6460,2010,pp. 326–336. [37]W.S.Massey,AlgebraicTopology:AnIntroduction,Harcourt,BraceandWorld,1967.

[38]J.Nešetˇril,Homomorphismsofderivativegraphs,DiscreteMath.1(1971)257–268.

[39]A.Pekeˇc,F.S.Roberts,Theroleassignmentmodelnearlyfitsmostsocialnetworks,Math.SocialSci.41(2001)275–293. [40]F.S.Roberts,L.Sheng,Howhardisittodetermineifagraphhasa2-roleassignment?,Networks37(2001)67–73.