We consider homomorphisms of signed graphs from a computational perspective. In particular, we study the list homomorphism problem seeking a homomorphism of an input signed graph $(G,\sigma)$, equipped with lists $L(v) \subseteq V(H), v \in V(G)$, of allowed images, to a fixed target signed graph $(H,\pi)$. The complexity of the similar homomorphism problem without lists (corresponding to all lists being $L(v)=V(H)$) has been previously classified by Brewster and Siggers, but the list version remains open and appears difficult. We illustrate this difficulty by classifying the complexity of the problem when $H$ is a tree (with possible loops). The tools we develop will be useful for classifications of other classes of signed graphs, and in a future companion paper we will illustrate this by using them to classify the complexity for certain irreflexive signed graphs. The structure of the signed trees in the polynomial cases is interesting, suggesting that the class of general signed graphs for which the problems are polynomial may have nice structure, analogous to the so-called bi-arc graphs (which characterized the polynomial cases of list homomorphisms to unsigned graphs).

翻译：我们从计算角度考虑带符号图的同态问题。特别地，我们研究列表同态问题，该问题寻求将输入带符号图 $(G,\sigma)$（配备允许像的列表 $L(v) \subseteq V(H), v \in V(G)$）映射到固定目标带符号图 $(H,\pi)$ 的同态。此前，Brewster 和 Siggers 已经分类了无列表版本（对应所有列表为 $L(v)=V(H)$）的类似同态问题的复杂性，但列表版本仍然开放且似乎具有挑战性。我们通过分类当 $H$ 为树（可能含自环）时问题的复杂性来阐述这一困难。我们开发的工具将有助于其他类别带符号图的分类，在未来的配套论文中，我们将通过使用这些工具对某些无自反带符号图进行复杂性分类来说明这一点。多项式情形下带符号树的结构令人感兴趣，这表明问题为多项式的广义带符号图类可能具有良好结构，类似于所谓的双弧图（它刻画了到无符号图的列表同态的多项式情形）。