A homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$. For a fixed graph $H$, in the list homomorphism problem, denoted by LHom($H$), we are given a graph $G$, whose every vertex $v$ is equipped with a list $L(v) \subseteq V(H)$. We ask if there exists a homomorphism $f$ from $G$ to $H$, in which $f(v) \in L(v)$ for every $v \in V(G)$. Feder, Hell, and Huang [JGT~2003] proved that LHom($H$) is polynomial time-solvable if $H$ is a bi-arc-graph, and NP-complete otherwise. We are interested in the complexity of the LHom($H$) problem in graphs excluding a copy of some fixed graph $F$ as an induced subgraph. It is known that if $F$ is connected and is not a path nor a subdivided claw, then for every non-bi-arc graph the LHom($H$) problem is NP-complete and cannot be solved in subexponential time, unless the ETH fails. We consider the remaining cases for connected graphs $F$. If $F$ is a path, we exhibit a full dichotomy. We define a class called predacious graphs and show that if $H$ is not predacious, then for every fixed $t$ the LHom($H$) problem can be solved in quasi-polynomial time in $P_t$-free graphs. On the other hand, if $H$ is predacious, then there exists $t$, such that LHom($H$) cannot be solved in subexponential time in $P_t$-free graphs. If $F$ is a subdivided claw, we show a full dichotomy in two important cases: for $H$ being irreflexive (i.e., with no loops), and for $H$ being reflexive (i.e., where every vertex has a loop). Unless the ETH fails, for irreflexive $H$ the LHom($H$) problem can be solved in subexponential time in graphs excluding a fixed subdivided claw if and only if $H$ is non-predacious and triangle-free. If $H$ is reflexive, then LHom($H$) cannot be solved in subexponential time whenever $H$ is not a bi-arc graph.
翻译:从GG美元到GG美元,我们要求从GG美元到GH美元, 将OFP美元保存在V(G)美元到美元(H)美元。对于固定的GP $H美元,在LHom(H$)表示的单调问题中,我们得到的是G$,每个G$都配有一份名单$L(v)\subseteq V(H)美元。我们问是否有一个PFP美元到$(HF$)的平面图,其中美元(V)美元到美元(H)。对于每1美元(G)美元(HF),美元(HFG$)的平面图。如果LHM(H$)是平面的平面图,如果平面的平面图是平面的,我们对LHOF美元(H)问题感兴趣,如果平面的平面的平面图不是平面的平面,除非平面的平面是平面的平面的平面,除非是平面的平面的。