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Complexity of planar signed graph homomorphisms to cycles

Abstract : We study homomorphism problems of signed graphs from a computational point of view. A signed graph is an undirected graph where each edge is given a sign, positive or negative. An important concept when studying signed graphs is the operation of switching at a vertex, which is to change the sign of each incident edge. A homomorphism of a graph is a vertex-mapping that preserves the adjacencies; in the case of signed graphs, we also preserve the edge-signs. Special homomorphisms of signed graphs, called s-homomorphisms, have been studied. In an s-homomorphism, we allow, before the mapping, to perform any number of switchings on the source signed graph. The concept of s-homomorphisms has been extensively studied, and a full complexity classification (polynomial or NP-complete) for s-homomorphism to a fixed target signed graph has recently been obtained. Nevertheless, such a dichotomy is not known when we restrict the input graph to be planar, not even for non-signed graph homomorphisms. We show that deciding whether a (non-signed) planar graph admits a homomorphism to the square C 2 t of a cycle with t 6, or to the circular clique K 4t/(2t−1) with t 2, are NP-complete problems. We use these results to show that deciding whether a planar signed graph admits an s-homomorphism to an unbalanced even cycle is NP-complete. (A cycle is unbalanced if it has an odd number of negative edges). We deduce a complete complexity dichotomy for the planar s-homomorphism problem with any signed cycle as a target. We also study further restrictions involving the maximum degree and the girth of the input signed graph. We prove that planar s-homomorphism problems to signed cycles remain NP-complete even for inputs of maximum degree 3 (except for the case of unbalanced 4-cycles, for which we show this for maximum degree 4). We also show that for a given integer g, the problem for signed bipartite planar inputs of girth g is either trivial or NP-complete.
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Submitted on : Monday, November 16, 2020 - 10:17:02 AM
Last modification on : Tuesday, October 12, 2021 - 5:20:41 PM
Long-term archiving on: : Wednesday, February 17, 2021 - 6:30:26 PM

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François Dross, Florent Foucaud, Valia Mitsou, Pascal Ochem, Théo Pierron. Complexity of planar signed graph homomorphisms to cycles. Discrete Applied Mathematics, Elsevier, 2020, 284, pp.166-178. ⟨10.1016/j.dam.2020.03.029⟩. ⟨hal-02990576⟩

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