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) is a graph G where each edge is given a sign, positive or negative; Σ⊆E(G) denotes the set of negative edges. Thus, (G,Σ) is a 2-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) to a signed graph (H,Π): ec-homomorphisms and s-homomorphisms. Each is a standard graph homomorphism of G to H 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 G.