Min Orderings and List Homomorphism Dichotomies for Signed and Unsigned Graphs

List Homomorphisms to Separable Signed Graphs

“…Extrapolating from these results, Kim and Siggers [25] conjectured that for all weakly balanced irreflexive signed graphs, the list homomorphism problem is polynomial-time solvable if there is special min ordering, and otherwise there is a chain or an invertible pair and the problem is NP-complete. We have proved this conjecture in [6]. Theorem 6.…”

“…For lists homomorphisms of signed graphs, there are several special cases where the complexity have been classified. These include signed graphs without bicoloured edges [5], signed trees with possible loops [4], and weakly balanced reflexive and irreflexive signed graphs [6,7,25]. We first introduce the relevant structures used to prove NP-completeness results and used to construct polynomial time algorithms.…”

“…Theorem 6. [6] For a weakly balanced irreflexive signed graph H, the list homomorphism problem is polynomial-time solvable if H has a special min ordering; otherwise, H contains a chain or an invertible pair and the problem is NP-complete.…”

“…A short preliminary version of this paper (containing only a small subset of the proofs) has appeared in the conference [1], prior to our paper [6]. In this expanded journal version we provide all proofs, taking advantage of now having Theorem 6 to simplify some of the arguments.…”

“…It is easy to check that the special property holds in all cases, and that it also holds for all black vertices. Thus our ordering is a special min ordering and we can use the result from [6,25] which asserts that for weakly balanced signed graphs the existence of a special min ordering ensures the existence of a polynomial-time algorithm.…”