The complexity of signed graph and edge-coloured graph homomorphisms

We study homomorphism problems of signed graphs from a computational point of view. A signed graph (G,Σ)(G,Σ) is a graph GG where each edge is given a sign, positive or negative; Σ⊆E(G)Σ⊆E(G) denotes the set of negative edges. Thus, (G,Σ)(G,Σ) is a 22-edge-coloured graph with the property that the edge-colours, {+,−}{+,−}, form a group under multiplication. Central to the study of signed graphs is the operation of switching at a vertex, that results in changing the sign of each incident edge. We study two types of homomorphisms of a signed graph (G,Σ)(G,Σ) to a signed graph (H,Π)(H,Π): ec-homomorphisms and s-homomorphisms. Each is a standard graph homomorphism of GG to HH with some additional constraint. In the former, edge-signs are preserved. In the latter, edge-signs are preserved after the switching operation has been applied to a subset of vertices of GG.We prove a dichotomy theorem for s-homomorphism problems for a large class of (fixed) target signed graphs (H,Π)(H,Π). Specifically, as long as (H,Π)(H,Π) does not contain a negative (respectively a positive) loop, the problem is polynomial-time solvable if the core of (H,Π)(H,Π) has at most two edges, and is NP-complete otherwise. (Note that this covers all simple signed graphs.) The same dichotomy holds if (H,Π)(H,Π) has no negative digons, and we conjecture that it holds always. In our proofs, we reduce s-homomorphism problems to certain ec-homomorphism problems, for which we are able to show a dichotomy. In contrast, we prove that a dichotomy theorem for ec-homomorphism problems (even when restricted to bipartite target signed graphs) would settle the dichotomy conjecture of Feder and Vardi.