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 $\widehat{H}$, the list homomorphism problem asks whether an input signed graph $\widehat{G}$ with lists $L(v) \subseteq V(\widehat{H}), v \in V(\widehat{G}),$ admits a homomorphism $f$ to $\widehat{H}$ with all $f(v) \in L(v), v \in V(\widehat{G})$. Usually, 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. We confirm their conjecture for reflexive and irreflexive signed graphs; this generalizes previous results on weakly balanced signed trees, and weakly balanced separable signed graphs \cite{separable,trees}. In the reflexive case, the result was first presented in \cite{KS}, with the proof using some of our results included in this paper. In fact, here we present our full proof, as an alternative to the proof in \cite{KS}. In particular, we provide direct polynomial algorithms where previously algorithms relied on general dichotomy theorems. The irreflexive results are new, and their proof depends on first deriving a theorem on extensions of min orderings of (unsigned) bipartite graphs, which is interesting on its own. [shortened, full abstract in PDF]

翻译：CSP二分猜想最近已被证明，但仍有许多其他二分性问题悬而未决，包括符号图列表同态问题的二分分类。符号图自然出现在多种语境中，例如在不可定向曲面上嵌入图的无处为零流问题。对于固定的符号图 $\widehat{H}$，列表同态问题询问：输入符号图 $\widehat{G}$ 及列表 $L(v) \subseteq V(\widehat{H}), v \in V(\widehat{G})$，是否存在同态 $f$ 从 $\widehat{G}$ 到 $\widehat{H}$，使得对所有 $v \in V(\widehat{G})$ 有 $f(v) \in L(v)$。通常，列表同态的二分分类比同态更易获得，但在符号图背景下，即使同态问题的复杂度分类已知，列表同态问题复杂度的结构分类甚至尚未提出猜想。Kim 和 Siggers 在“弱平衡”符号图的特例中推测了一种结构分类。我们证实了他们对自反与反身符号图的猜想；这推广了此前关于弱平衡符号树及弱平衡可分离符号图的研究结果。在自反情况下，该结果首次发表于文献 [KS]，其证明使用了本文包含的部分结果。实际上，我们在此给出完整证明，作为文献 [KS] 中证明的替代方案。特别地，我们提供了直接的多项式算法，而此前算法依赖于通用二分定理。反身情况结果是全新的，其证明首先需要推导关于（无符号）二部图最小序扩展的定理，这本身具有独立意义。[已缩短，完整摘要见PDF]