Towards A Dichotomy for the List Switch Homomorphism Problem for Signed Graphs

We make advances towards a structural characterisation of the signed graphs H for which the list switch H-colouring problem List-S-Hom(H)can be solved in polynomial time. We conjecture two different characterisations, the second refining the first, in the case that the graph H can be switched to a graph in which every negative edge is also positive. Using a recent proof of the first characterisations for reflexive signed graphs, by Bok et. al., we prove the second characterisation for reflexive signed graphs. We also provide several tools for reducing the problem to the bipartite case, and prove a full complexity dichotomy for a related problem.