eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
30:1
30:15
10.4230/LIPIcs.ISAAC.2022.30
article
List Locally Surjective Homomorphisms in Hereditary Graph Classes
Dvořák, Pavel
1
2
https://orcid.org/0000-0002-6838-1538
Masařík, Tomáš
3
https://orcid.org/0000-0001-8524-4036
Novotná, Jana
3
4
https://orcid.org/0000-0002-7955-4692
Krawczyk, Monika
5
Rzążewski, Paweł
5
3
https://orcid.org/0000-0001-7696-3848
Żuk, Aneta
5
Tata Institute of Fundamental Research, Mumbai, India
Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
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
Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland
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).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.30/LIPIcs.ISAAC.2022.30.pdf
Homomorphism
Hereditary graphs
Subexponential-time algorithms