A signed graph is a graph in which each edge has a positive or negative sign. Calling two graphs switch equivalent if one can get from one to the other by the iteration of the local action of switching all signs on edges incident to a given vertex, we say that there is a switch homomorphism from a signed graph $G$ to a signed graph $H$ if there is a sign preserving homomorphism from $G’$ to $H$ for some graph $G’$ that is switch equivalent to $G$. By reductions to CSP this problem, and its list version, are known to be either polynomial time solvable or NP-complete, depending on $H$. Recently those signed graphs $H$ for which the switch homomorphism problem is in $P$ were characterised. Such a characterisation is yet unknown for the list version of the problem.
We talk about recent work towards such a characterisation and about how these problems fit in with bigger questions that still remain around the recent CSP dichotomy theorem.