The complexity of signed graph and edge-coloured graph homomorphisms

The goal of this work is to study homomorphism problems (from a computational point of view) on two superclasses of graphs: 2-edge-coloured graphs and signed graphs. On the one hand, we consider the H-Colouring problem when H is a 2-edge-coloured graph, and we show that a dichotomy theorem would imply the dichotomy conjecture of Feder and Vardi. On the other hand, we prove a dichotomy theorem for the (H,Π)-Colouring problem for a large class of signed graphs (H,Π). Specifically, as long as (H,Π) does not contain a negative (respectively a positive) loop, the problem is polynomial-time solvable if the core of (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) has no digons, and we conjecture that it holds always.