1,713 research outputs found

    Forbidden Directed Minors and Kelly-width

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    Partial 1-trees are undirected graphs of treewidth at most one. Similarly, partial 1-DAGs are directed graphs of KellyWidth at most two. It is well-known that an undirected graph is a partial 1-tree if and only if it has no K_3 minor. In this paper, we generalize this characterization to partial 1-DAGs. We show that partial 1-DAGs are characterized by three forbidden directed minors, K_3, N_4 and M_5

    On some special classes of contact B0B_0-VPG graphs

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    A graph GG is a B0B_0-VPG graph if one can associate a path on a rectangular grid with each vertex such that two vertices are adjacent if and only if the corresponding paths intersect at at least one grid-point. A graph GG is a contact B0B_0-VPG graph if it is a B0B_0-VPG graph admitting a representation with no two paths crossing and no two paths sharing an edge of the grid. In this paper, we present a minimal forbidden induced subgraph characterisation of contact B0B_0-VPG graphs within four special graph classes: chordal graphs, tree-cographs, P4P_4-tidy graphs and P5P_5-free graphs. Moreover, we present a polynomial-time algorithm for recognising chordal contact B0B_0-VPG graphs.Comment: 34 pages, 15 figure

    Graph quasivarieties

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    Introduced by C. R. Shallon in 1979, graph algebras establish a useful connection between graph theory and universal algebra. This makes it possible to investigate graph varieties and graph quasivarieties, i.e., classes of graphs described by identities or quasi-identities. In this paper, graph quasivarieties are characterized as classes of graphs closed under directed unions of isomorphic copies of finite strong pointed subproducts.Comment: 15 page

    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
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