List Locally Surjective Homomorphisms in Hereditary Graph Classes

Abstract

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 for F being a bounded-degree forest, unless the ETH fails. The initial study reveals that a natural subfamily of boundeddegree forests F , that might lead to some tractability results, is the family 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 ∈ {P3, C4} are the only connected ones that allow for a subexponential-time algorithm in F -free graphs for every F ∈ S (unless the ETH fails).