Complexity of the list homomorphism problem in hereditary graph classes
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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) ⊆ V(H). We ask if there exists a homomorphism f from G to H, in which f(v) ∈ L(v) for every v ∈ 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.
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