The CSP dichotomy conjecture has been recently established, but a number of other dichotomy questions remain open, including the dichotomy classification of list homomorphism problems for signed graphs. Signed graphs arise naturally in many contexts, including for instance nowhere-zero flows for graphs embedded in non-orientable surfaces. For a fixed signed graph H, the list homomorphism problem asks whether an input signed graph G with listsUsually, a dichotomy classification is easier to obtain for list homomorphisms than for homomorphisms, but in the context of signed graphs a structural classification of the complexity of list homomorphism problems has not even been conjectured, even though the classification of the complexity of homomorphism problems is known. Kim and Siggers have conjectured a structural classification in the special case of "weakly balanced" signed graphs, and proved it for reflexive signed graphs. We confirm the conjecture for irreflexive signed graphs; this generalizes previous results on weakly balanced signed trees, and weakly balanced separable signed graphs. Our proof depends on first deriving a new result on extensions of min orderings of (unsigned) bipartite graphs.
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