665 research outputs found

    Testing List H-Homomorphisms

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    Let HH be an undirected graph. In the List HH-Homomorphism Problem, given an undirected graph GG with a list constraint L(v)⊆V(H)L(v) \subseteq V(H) for each variable v∈V(G)v \in V(G), the objective is to find a list HH-homomorphism f:V(G)→V(H)f:V(G) \to V(H), that is, f(v)∈L(v)f(v) \in L(v) for every v∈V(G)v \in V(G) and (f(u),f(v))∈E(H)(f(u),f(v)) \in E(H) whenever (u,v)∈E(G)(u,v) \in E(G). We consider the following problem: given a map f:V(G)→V(H)f:V(G) \to V(H) as an oracle access, the objective is to decide with high probability whether ff is a list HH-homomorphism or \textit{far} from any list HH-homomorphisms. The efficiency of an algorithm is measured by the number of accesses to ff. In this paper, we classify graphs HH with respect to the query complexity for testing list HH-homomorphisms and show the following trichotomy holds: (i) List HH-homomorphisms are testable with a constant number of queries if and only if HH is a reflexive complete graph or an irreflexive complete bipartite graph. (ii) List HH-homomorphisms are testable with a sublinear number of queries if and only if HH is a bi-arc graph. (iii) Testing list HH-homomorphisms requires a linear number of queries if HH is not a bi-arc graph

    Uniquely D-colourable digraphs with large girth

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    Let C and D be digraphs. A mapping f:V(D)→V(C)f:V(D)\to V(C) is a C-colouring if for every arc uvuv of D, either f(u)f(v)f(u)f(v) is an arc of C or f(u)=f(v)f(u)=f(v), and the preimage of every vertex of C induces an acyclic subdigraph in D. We say that D is C-colourable if it admits a C-colouring and that D is uniquely C-colourable if it is surjectively C-colourable and any two C-colourings of D differ by an automorphism of C. We prove that if a digraph D is not C-colourable, then there exist digraphs of arbitrarily large girth that are D-colourable but not C-colourable. Moreover, for every digraph D that is uniquely D-colourable, there exists a uniquely D-colourable digraph of arbitrarily large girth. In particular, this implies that for every rational number r≥1r\geq 1, there are uniquely circularly r-colourable digraphs with arbitrarily large girth.Comment: 21 pages, 0 figures To be published in Canadian Journal of Mathematic

    Minimum Cost Homomorphisms to Reflexive Digraphs

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    For digraphs GG and HH, a homomorphism of GG to HH is a mapping $f:\ V(G)\dom V(H)suchthat such that uv\in A(G)implies implies f(u)f(v)\in A(H).Ifmoreovereachvertex. If moreover each vertex u \in V(G)isassociatedwithcosts is associated with costs c_i(u), i \in V(H),thenthecostofahomomorphism, then the cost of a homomorphism fis is \sum_{u\in V(G)}c_{f(u)}(u).Foreachfixeddigraph. For each fixed digraph H, the {\em minimum cost homomorphism problem} for H,denotedMinHOM(, denoted MinHOM(H),isthefollowingproblem.Givenaninputdigraph), is the following problem. Given an input digraph G,togetherwithcosts, together with costs c_i(u),, u\in V(G),, i\in V(H),andaninteger, and an integer k,decideif, decide if Gadmitsahomomorphismto admits a homomorphism to Hofcostnotexceeding of cost not exceeding k. We focus on the minimum cost homomorphism problem for {\em reflexive} digraphs H(everyvertexof (every vertex of Hhasaloop).ItisknownthattheproblemMinHOM( has a loop). It is known that the problem MinHOM(H)ispolynomialtimesolvableifthedigraph) is polynomial time solvable if the digraph H has a {\em Min-Max ordering}, i.e., if its vertices can be linearly ordered by <sothat so that i<j, s<rand and ir, js \in A(H)implythat imply that is \in A(H)and and jr \in A(H).WegiveaforbiddeninducedsubgraphcharacterizationofreflexivedigraphswithaMin−Maxordering;ourcharacterizationimpliesapolynomialtimetestfortheexistenceofaMin−Maxordering.Usingthischaracterization,weshowthatforareflexivedigraph. We give a forbidden induced subgraph characterization of reflexive digraphs with a Min-Max ordering; our characterization implies a polynomial time test for the existence of a Min-Max ordering. Using this characterization, we show that for a reflexive digraph H$ which does not admit a Min-Max ordering, the minimum cost homomorphism problem is NP-complete. Thus we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for reflexive digraphs

    The Complexity of Surjective Homomorphism Problems -- a Survey

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    We survey known results about the complexity of surjective homomorphism problems, studied in the context of related problems in the literature such as list homomorphism, retraction and compaction. In comparison with these problems, surjective homomorphism problems seem to be harder to classify and we examine especially three concrete problems that have arisen from the literature, two of which remain of open complexity
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