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 and reflexive signed graphs. 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 semi-balanced irreflexive signed graphs; we have proved the conjecture in another paper, and the present results add structural information to that topic.

翻译：符号图上列表同态问题的复杂性似乎难以分类。现有结果主要关注特殊类型的符号图，如树和自反符号图。正如Kim和Siggers近期论文所指出的，非自反符号图在某种意义上构成了该问题的核心。我们聚焦于一类特殊的非自反符号图——其中单色边构成生成路径或圈，并将其称为可分离符号图。我们分类了面向这些可分离符号图的列表同态复杂性，认为这类符号图将对非自反情形的一般解法发挥重要作用。我们还将其与Kim和Siggers关于半平衡非自反符号图特殊情形的猜想建立关联；我们已在另一篇论文中证明了该猜想，而本文结果为该课题补充了结构信息。