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List Locally Surjective Homomorphisms in Hereditary Graph Classes

Authors Pavel Dvořák , Tomáš Masařík , Jana Novotná , Monika Krawczyk, Paweł Rzążewski , Aneta Żuk

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Author Details

Pavel Dvořák
  • Tata Institute of Fundamental Research, Mumbai, India
  • Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Tomáš Masařík
  • Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
Jana Novotná
  • Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Monika Krawczyk
  • Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland
Paweł Rzążewski
  • Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland
  • Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
Aneta Żuk
  • Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland

Funding

  • Dvořák, Pavel: Partially supported by EPSRC New Investigator Award EP/V010611/1 and by Czech Science Foundation GAČR grant #22-14872O.
  • Masařík, Tomáš: Received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme Grant Agreement 948057.
  • Novotná, Jana: Supported by SVV-2020–260578 and GAUK 384321 of Charles University.
  • Rzążewski, Paweł: Supported by Polish National Science Centre grant no. 2018/31/D/ST6/00062.

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Pavel Dvořák, Tomáš Masařík, Jana Novotná, Monika Krawczyk, Paweł Rzążewski, and Aneta Żuk. List Locally Surjective Homomorphisms in Hereditary Graph Classes. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ISAAC.2022.30

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 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).

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Graph algorithms
Keywords
  • Homomorphism
  • Hereditary graphs
  • Subexponential-time algorithms

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