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.We prove a dichotomy theorem for s-homomorphism problems for a large class of (fixed) target 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 negative digons, and we conjecture that it holds always. In our proofs, we reduce s-homomorphism problems to certain ec-homomorphism problems, for which we are able to show a dichotomy. In contrast, we prove that a dichotomy theorem for ec-homomorphism problems (even when restricted to bipartite target signed graphs) would settle the dichotomy conjecture of Feder and Vardi.

Let p and q be positive integers with p/q ≥ 2. The "reconfiguration problem" for circular colourings asks, given two (p, q)-colourings f and g of a graph G, is it possible to transform f into g by changing the colour of one vertex at a time such that every intermediate mapping is a (p, q)-colouring? We show that this problem can be solved in polynomial time for 2 ≤ p/q < 4 and that it is PSPACE-complete for p/q ≥ 4. This generalizes a known dichotomy theorem for reconfiguring classical graph colourings. As an application of the reconfiguration algorithm, we show that graphs with fewer than (k − 1)!/2 cycles of length divisible by k are k-colourable.

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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 [4] and reflexive signed graphs [25]. 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 weakly balanced irreflexive signed graphs; we have proved the conjecture in another paper, and the present results add structural information to that topic.

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