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List Locally Surjective Homomorphisms in Hereditary Graph Classes Dvořák, Pavel Masařík, Tomáš Novotná, Jana Krawczyk, Monika Rzążewski, Paweł Żuk, Aneta Homomorphism Hereditary graphs Subexponential-time algorithms 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 bounded-degree 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). Schloss Dagstuhl – Leibniz-Zentrum für Informatik Pavel Dvořák and Tomáš Masařík and Jana Novotná and Monika Krawczyk and Paweł Rzążewski and Aneta Żuk 2022 Is Part Of LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ISAAC.2022.30 urn:nbn:de:0030-drops-173154 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.30 eng https://creativecommons.org/licenses/by/4.0/legalcode