List homomorphism problems for signed graphs

We consider homomorphisms of signed graphs from a computational perspective. In particular, we study the list homomorphism problem seeking a homomorphism of an input signed graph $(G,\sigma)$, equipped with lists $L(v) \subseteq V(H), v \in V(G)$, of allowed images, to a fixed target signed graph $(H,\pi)$. The complexity of the similar homomorphism problem without lists (corresponding to all lists being $L(v)=V(H)$) has been previously classified by Brewster and Siggers, but the list version remains open and appears difficult. We illustrate this difficulty by classifying the complexity of the problem when $H$ is a tree (with possible loops). The tools we develop will be useful for classifications of other classes of signed graphs, and we illustrate this by classifying the complexity of irreflexive signed graphs in which the unicoloured edges form some simple structures, namely paths or cycles. The structure of the signed graphs in the polynomial cases is interesting, suggesting they may constitute a nice class of signed graphs analogous to the so-called bi-arc graphs (which characterize the polynomial cases of list homomorphisms to unsigned graphs).Comment: various changes + rewritten section on path- and cycle-separable graphs based on a new conference submission (split possible in future

Min orderings and list homomorphism dichotomies for signed and unsigned graphs

The CSP dichotomy conjecture has been recently established, but a number of other dichotomy questions remain open, including the dichotomy classification of list homomorphism problems for signed graphs. Signed graphs arise naturally in many contexts, including for instance nowhere-zero flows for graphs embedded in non-orientable surfaces. For a fixed signed graph $\widehat{H}$, the list homomorphism problem asks whether an input signed graph $\widehat{G}$ with lists $L(v) \subseteq V(\widehat{H}), v \in V(\widehat{G}),$ admits a homomorphism $f$ to $\widehat{H}$ with all $f(v) \in L(v), v \in V(\widehat{G})$. Usually, a dichotomy classification is easier to obtain for list homomorphisms than for homomorphisms, but in the context of signed graphs a structural classification of the complexity of list homomorphism problems has not even been conjectured, even though the classification of the complexity of homomorphism problems is known. Kim and Siggers have conjectured a structural classification in the special case of "weakly balanced" signed graphs. We confirm their conjecture for reflexive and irreflexive signed graphs; this generalizes previous results on weakly balanced signed trees, and weakly balanced separable signed graphs. In the reflexive case, the result was first presented in a paper of Kim and Siggers, where the proof relies on a result in this paper. The irreflexive result is new, and its proof depends on first deriving a theorem on extensions of min orderings of (unsigned) bipartite graphs, which is interesting on its own

Complexity of planar signed graph homomorphisms to cycles

We study homomorphism problems of signed graphs. A signed graph is an undirected graph where each edge is given a sign, positive or negative. An important concept for 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. This concept 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. 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 $C_t^2$ of a cycle with $t\ge 6$, or to the circular clique $K_{4t/(2t-1)}$ with $t\ge2$, 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.Comment: 17 pages, 10 figure