Locallyinjectivegraphhomomorphism:ListsguaranteedichotomyJir FialaandJanKratochvl1DepartmentofAppliedMathematicsandInstituteforTheoretical ComputerScience(ITI),FacultyofMathematicsandPhysics,CharlesUniversity,Malostranskenam.2/25, 11800Prague,CzechRepublic{fiala,honza}@kam.mff.cuni.czAbstract. WeprovethatintheListversion, theproblemof de-cidingtheexistenceof alocallyinjectivehomomorphismtoapa-rametergraphHperformsafulldichotomy.NamelyweshowthatitispolynomiallytimesolvableifeveryconnectedcomponentofHhasatmostonecycleandNP-completeotherwise.1 IntroductionWeconsider niteundirectedgraphs without loops or multipleedges. Arecentlyintensivelystudiednotion, for its algebraicmotivationandcon-nectionsaswell asbeinganaturalgeneralizationofgraphcoloring,isthenotionof graph homomorphisms. Ahomomorphismf : GHfromagraphGtoagraphHisanedge-preservingvertexmapping, i.e., amap-pingf:V (G)V (H)suchthatf(x)f(y)E(H)wheneverxyE(G).(For a recent monograph on graph homomorphisms the reader is referred to[14].) It follows from thedenition that the neighborhood of every vertex ismapped into the neighborhood of its image, formally f(NG(x)) NH(f(x))forallx V (G).Properties oftheserestricted mappings, local constraints,lead to the denition of locallyconstrainedhomomorphisms. The homomor-phismf is calledlocally injective(bijective, surjective, resp.) if for everyx V (G), the restrictedmapping f : NG(x) NH(f(x)) is injective(bijective, surjective, resp.). All threeof thesenotions havebeenstudiedon theirown with dierent motivations. Locally surjective homomorphismsSupportedby the Ministry of Education of the Czech Republic as project1M0021620808.1correspondtosocalledroleassignmentgraphsstudiedinsociological ap-plications [11], locally bijective ones correspond to graph covers well knownfrom topological graph theory [2, 15] and theory of local computation [1, 3].Locallyinjectivehomomorphisms arecloselyrelated togeneralizedL(2, 1)-labelingsofgraphsandtheFrequencyAssignmentProblem[8, 9].Fromthecomputational complexitypointof viewweareinterestedinthe decision problem if an input graph G allows a homomorphism of certaintypeintoaxedtargetgraphH. AstheseproblemsareparametrizedbythegraphH, weusethenotationH-Hom (when asking for theexistence ofany homomorphism), H-LIHom,H-LBHom and H-LSHom (when asking forlocallyinjective,bijectiveorsurjectivehomomorphisms, respectively).The complexity of H-Hom is fully understood and dichotomy was provedby Hell and Nesetril in [13]. The problem is polynomially solvable if His bi-partite andNP-complete otherwise. Thecomplexity ofH-LSHom was stud-ied in [17] and completed byFialaand Paulusma in [11]. Thisproblem (forconnected graph H) is solvable in polynomial time if H has at most 2 verticesandNP-completeotherwise.Severalpapershavebeendevotedtostudyingthecomplexityoflocallyinjectiveandbijectivehomomorphisms,butonlypartial results areknown[8, 10, 16]. Thoughconjecturedat least for thecaseoflocallybijectivehomomorphisms,thePolynomial/NP-completenessdichotomyhasbeenprovedinneitherof thelasttwocases.However,thisquestionis natural especiallyinviewof thefact that locallyconstrainedhomomorphismscanbeexpressedasConstraintSatisfactionProblem, forwhich the dichotomy was conjectured in the fully general case by Feder andVardy[6].The CSP view also suggests considering the List versions of the problems,since lists correspond to unary relations (cf. next section). The input to theListversionof ahomomorphismproblemisagraphGtogetherwithlistsL(u) V (H) of admissible targets for every vertex u V (G). The questionisifGallowsa(locallyinjective,bijectiveetc.)homomorphismf:G Hsuchthat f(u) L(u) for everyuV (G). Werefer totheseproblemsas List-H-Hom, List-H-LIHometc. Adeepresultof Hell etal. givesafullcharacterization of the case of general homomorphisms [5, 12]. The problemList-H-Hom is polynomially solvable for so called double circular arc graphsHandNP-completeotherwise.Settingthelists totheentirevertexset of thetarget graph, oneim-mediatelyseesthatH-HomList-H-Hom, H-LIHomList-H-LIHom, H-LBHom List-H-LBHom and H-LSHom List-H-LSHom. Thus in each casetheborderlinebetweenpolynomialandNP-completeinstances(dichotomy2assumed) of the List version will lie within the easy instances of the non-Listone. This is well seen in the above mentioned case of general homomorphismsand also in thecase of thelocallysurjective ones List-H-LSHom remainspolynomial forgraphsHwithatmosttwovertices. Thetroublewiththelocally injective and locally bijective homomorphisms is that the full charac-terization of thenon-List versions isnot known. Nevertheless, listsdo help!Thepurposeof thispaperistoshowthatinthecaseof locallyinjectivehomomorphisms, listsguarantee afulldichotomy.Theorem1. The List-H-LIHomproblemis solvable inlinear time if thegraphHcontainsatmostonecycleineachconnectedcomponent,anditisNP-completeotherwise.The paper is structured as follows. In the next section we quickly describethe connectionto the Constraint SatisfactionProblem. InSection3 wegivetheargumentforthepolynomial partof ourtheorem. Thetechnicalreductions for theNP-hardness part are presented in Section 4 and the proofissummarizedinSection5.Thenalsectioncontainsconcludingremarks.2 LocallyconstrainedhomomorphismsasCSPTheCSPisparametrizedbyaxedtemplateX=(X; S1, . . . , Sk), whereSisarerelationsonanitesetX,thearityofSibeingni, i = 1, 2, . . . , k.TheinputoftheX-CSPisastructureU=(U; R1, . . . , Rk), whereUisa(large) set and Ri is an ni-ary relation on U, for i = 1, 2, . . . , k. The questionis whether thereexists amapping(infact, astructural homomorphism)f : UXsuchthatforeveryi andeveryni-tuple(u1, . . . , uni)Uni,(u1, . . . , uni) Riimplies (f(u1), . . . , f(uni)) Si. Feder andVardy[6]conjecture that for everytemplate X, this problemis either polynomialtimesolvable orNP-complete. Thisis known for binary structures [19], andformanyspecialcases(cf.e.g.,Bulatovetal.[4]).Though in natural structures one tends to overlook unary relations, fromthepointof viewof theformal denitiontheyform afullycoherent part ofthepicture.Andtheycorrespond tolists.Aunaryrelationisjustasubsetof theground set. If a vertexu Ubelongs tounary relationsRi1, . . . , Rit,then the constraints given by the unary relations of X-CSP merely say thatf(u)

tj=1Sij, which is equivalent to settingthe list of admissible imagesofutoL(u) =

tj=1Sij.3The general homomorphism problemH-Hom is obviously a CSP problem the template is H itself, and the input structure is G(edges of both graphsare considered as symmetric binary relations). We will show that also locallyinjectiveandbijectivehomomorphismscanbeexpressedasCSP. GivenagraphHwithhvertices,setD = {(x, y)| x = y V (H)},Di= {(x)| degH(x) = i, x V (H)}, i = 0, 1, . . . , h 1.Here Dis the symmetric binaryrelationcontainingall pairs of distinctvertices andDisare unaryrelations controlling thedegrees. WederivethefollowingtemplatesfromH:LH= (V (H); S1= E(H), S2= D)andBH= (V (H); S1= E(H), S2= D, S3+i= Di, i = 0, . . . , h 1).Observation1For every graph H, H-LIHom LH-CSP and H-LBHom BH-CSP.Proof. For an input graph G, dene U = (V (G); R1= E(G), R2={(x, y)|; x=y z : xz, yz E(G)}). ThenUis afeasibleinstanceforLH-CSPifandonlyif GallowsalocallyinjectivehomomorphismintoHtherelationsR1andS1control thatacandidatevertexmappingisagraph homomorphism, andR2withS2control thelocal injectivity.Forthelocallybijectivecase, justaddR3+i={uV (G)| degG(u)=i}foreachi = 0, 1, . . . , h 1,toensurethatthismappingislocallybijectiveoneveryneighborhood.Observe, however, that this does not mean that the Feder-Vardy conjec-ture would implydichotomy forH-LIHom orH-LBHom. Our observation isusefulonlywhenthecorrespondingCSPproblemispolynomiallysolvable,whichis unfortunatelyonlyseldom(whenever Hhas at least threever-tices, 3-COLORABILITYLH-CSPBH-CSPandbothproblemsareNP-complete). Thepointisthattheinputs of LH-CSP(orBH-CSP)de-rived from G are not arbitrary. In view of this becomes the fact that addinglistsendorsesdichotomyonList-H-LIHomevenmoreinteresting.4PkHClCkGFig. 1.ThepolynomialinstancesfortheList-H-LIHomproblem.The2lpossibil-itiesof of mappingof theemphasizedCkandPkareafterwardsveriedforthependingtrees.3 Thepolynomial caseLemma1. ForanyconnectedgraphHcontainingat most onecycle, theList-H-LIHomproblemissolvableinlineartime.Proof. Without loss of generality we may assume that theinput graph G isconnected,sinceotherwisewecantreateachcomponentseparately.If His a tree, then any connected graph G that allows a locally injectivehomomorphismGHisasubtreeof H. SinceHisxed, thegraphGitself must have a bounded number of vertices and the problem H-LIHom issolvableinconstant timeevenwithlistsbeingincorporated.Let Hhave exactly one cycle. We rst note that the number of connectedgraphsGofdiameteratmost2 diam(H)thatallowalocallyinjectiveho-momorphism toHisbounded(each such graph hasat most 1 +2 diam(H)vertices, where is themaximumvertexdegreeof H). Hence, for suchinstances,theList-H-LIHomproblemcanbedecidedinconstanttime.For the rest of the proof suppose that diam(G) > 2 diam(H). Denote byCltheuniquecycleofH(consultFig3).Wedistinguishtwocases:Gcontainsacycle, sayCk: ThenthiscyclemustbemappedontoCl.Thismighthappeninatmost2l ways.Someofthese2l waysmaybefurther excluded by the list constraints. If we x a mappingCk Cl, itremains to decide whether this mapping can be extended to the remain-ing vertices of G. For every vertex u of Ck, we can solve this question in5constant time, since this is equivalent to the treeList-H-LIHom problem.(Nowtheinstanceisthecomponentof G \ ECkcontaininguandtheparameter is the component of H\ EClcontaining the image ofu. Botharetrees.)Gis a tree: Anecessary condition for Gto allowa locally injec-tive homomorphismto His that Gcontains a path Pkof lengthk = diam(G)2 diam(H) such that the components of G\EPkstemmingfrom theinnerverticesofPkmaplocallyinjectivelytothecomponentsof H\ECland the two trees of diameter diam(H) hanging on the terminiofPkmaplocallyinjectivelytoH.WerstndPkbydiam(H)manyiterationsof peelingoverticesof degreeone. Then, asintheabovecase,wetryall 2l possibilitiesof alocallyinjectivemappingPkClandexcludethosewhichdonotsatisfythelistconstraints.Finallyforeach suchpartialmapping,wecheckifitcan beextendedtotheentireGbysolvingatmostk + 1List-H-LIHomproblemsofconstantsize.4 AuxiliaryNP-hardnessreductionsIn this section we showNP-hardness of theList-H-LIHom problem for threebasic types of graphs depictedinFig. 1(theyare informallycalledtheTheta graph, the Flower graph and the Weight graph). The problem we useforourNP-hardnessreductionsisEdge-precoloringextension.Ithasbeen shownNP-complete even when restricted to cubic bipartite graphs [7].Theinputofthisvariantconsistsofacubicbipartitegraphtogetherwithapartial coloringofitsedgesbythreecolorssayred, blueandwhite1.Thequestioniswhetherthispartialcoloringcanbeextendedtotheentireedgesetsuchthateachvertexisincidentwithedgesofallthreecolors.Forpositiveintegersa, b, c, whereb, c 2, let(a, b, c)bethegraph ona+b +c 1 vertices consisting of two vertices of degree three connected bypathsoflengthsa,b,andc(cf.Fig.1topleft).Lemma2. Forarbitrarypositiveintegersa, b, c, whereb, c2, the List-(a, b, c)-LIHomproblemisNP-hard.Proof. If a=b=c2, wecanreducetheEdge-precoloringexten-sionproblemdirectly. LetGbetheinstanceof theEdge-precoloringextension problem. We replace each edge e E(G) by a path Peof length1ThefavoritetricolorforCzechsaswellasforseveralothernations.6s1sb1t1tc1....... . .y zra1r1B(a, b, c)v wm1n1o1oc1ma1nb1xp1pa1q1qb1. . .. . .. . .......F(a, b)(a, b, c)Fig. 2.Thebasicthreetypesofgraphs.a. Wefurtherassociatecolorsred, blueandwhitewiththeinnerverticesofthethreepathsof(a, a, a)insuchawaythatverticesoftherstpathrepresentcolorred, of thesecondpathcolorblue, andthoseof thethirdone the white color. If the edgee was precolored by a color , we assign theverticesofPelistsconsistingofvertices ofthepathin(a, a, a)represent-ingthecolor. Fortheotherverticesweletthelistsbethewholevertexsetof(a, a, a).Straightforwardly, proper3-edgecoloringsofGareinone-to-onecorre-spondencewithvalidlocallyinjectivehomomorphismsto(a, a, a). (Suchhomomorphismsmustinfactbelocallybijective.)Thelocalconditionsonvertices of degree threerepresent thecondition that thethreecolors on theedgesoftheoriginal graphmustbedistinct.For the caseof ab c, a