遗传图类中的列表局部满射同态
List Locally Surjective Homomorphisms in Hereditary Graph Classes
AI总结：
研究遗传图类中列表局部满射同态问题的复杂性，证明除非F是有界度森林或ETH失败否则无次指数算法，并对特定森林族给出二分定理。
摘要：
从图$G$到图$H$的局部满射同态是一个从$V(G)$到$V(H)$的保边映射，且在$G$中每个顶点的邻域内是满射的。在列表局部满射同态问题（记为LLSHom($H$)）中，图$H$是固定的，实例由一个图$G$组成，$G$的每个顶点都配备了$V(H)$的一个子集，称为列表。我们询问是否存在从$G$到$H$的局部满射同态，使得$G$的每个顶点都被映射到其列表中的某个顶点。在本文中，我们研究了LLSHom($H$)问题在$F$-free图（即排除固定图$F$作为导出子图的图）中的复杂性。我们旨在探讨对于哪些图对$(H,F)$，该问题可以在次指数时间内求解。我们证明，对于所有在一般图中该问题为NP困难的图$H$，除非$F$是有界度森林或ETH失败，否则该问题在$F$-free图中无法在次指数时间内求解。初步研究表明，可能导致一些可处理结果的有界度森林$F$的一个自然子族是由每个分支最多有三个叶子的森林组成的族$\mathcal S$。在这种情况下，我们展示了以下二分定理：除了在一般图中多项式时间可解的情况外，图$H \in \{P_3,C_4\}$是仅有的连通图，它们对于每个$F \in \mathcal S$在$F$-free图中允许次指数时间算法（除非ETH失败）。
英文摘要：
A locally surjective homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$ that is surjective in the neighborhood of each vertex in $G$. In the list locally surjective homomorphism problem, denoted by LLSHom($H$), the graph $H$ is fixed and the instance consists of a graph $G$ whose every vertex is equipped with a subset of $V(H)$, called list. We ask for the existence of a locally surjective homomorphism from $G$ to $H$, where every vertex of $G$ is mapped to a vertex from its list. In this paper, we study the complexity of the LLSHom($H$) problem in $F$-free graphs, i.e., graphs that exclude a fixed graph $F$ as an induced subgraph. We aim to understand for which pairs $(H,F)$ the problem can be solved in subexponential time. We show that for all graphs $H$, for which the problem is NP-hard in general graphs, it cannot be solved in subexponential time in $F$-free graphs unless $F$ is a bounded-degree forest or the ETH fails. The initial study reveals that a natural subfamily of bounded-degree forests $F$ that might lead to some tractability results is the family $\mathcal S$ consisting of forests whose every component has at most three leaves. In this case, we exhibit the following dichotomy theorem: besides the cases that are polynomial-time solvable in general graphs, the graphs $H \in \{P_3,C_4\}$ are the only connected ones that allow for a subexponential-time algorithm in $F$-free graphs for every $F \in \mathcal S$ (unless the ETH fails).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
