Min Orderings and List Homomorphism Dichotomies for Graphs and Signed Graphs

Since the CSP dichotomy conjecture has been established, a number of other dichotomy questions have attracted interest, including one for list homomorphism problems of signed graphs. Signed graphs arise naturally in many contexts, including for instance nowhere-zero flows for graphs embedded in non-orientable surfaces. The dichotomy classification is known for homomorphisms without list restrictions, so it is surprising that it is not known, or even conjectured, if lists are present since this usually makes the classifications easier to obtain. There is however a conjectured classification, due to Kim and Siggers, in the special case of “semi-balanced” signed graphs. These authors confirmed their conjecture for the class of reflexive signed graphs. As our main result we verify the conjecture for irreflexive signed graphs. For this purpose, we prove an extension result for two-directional ray graphs which is of independent interest and which leads to an analogous extension result for interval graphs. Moreover, we offer an alternative proof for the class of reflexive signed graphs, and a direct polynomial-time algorithm in the polynomial cases where the previous algorithms used algebraic methods of general CSP dichotomy theorems. For both reflexive and irreflexive cases the dichotomy classification depends on a result linking the absence of certain structures to the existence of a special ordering. The structures are used to prove the NP-completeness and the ordering is used to design polynomial algorithms.