  The complexity of the list homomorphism problem for signed graphs appears
difficult to classify. Existing results focus on special classes of signed
graphs, such as trees and reflexive signed graphs. Irreflexive signed graphs
are in a certain sense the heart of the problem, as noted by a recent paper of
Kim and Siggers. We focus on a special class of irreflexive signed graphs,
namely those in which the unicoloured edges form a spanning path or cycle,
which we call separable signed graphs. We classify the complexity of list
homomorphisms to these separable signed graphs; we believe that these signed
graphs will play an important role for the general resolution of the
irreflexive case. We also relate our results to a conjecture of Kim and Siggers
concerning the special case of semi-balanced irreflexive signed graphs; we have
proved the conjecture in another paper, and the present results add structural
information to that topic.
