Abstract
A locally surjective homomorphism from a graph G to a graph H is an edgepreserving 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 Ffree 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 NPhard in general graphs, it cannot be solved in subexponential time in Ffree graphs for F being a boundeddegree 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 polynomialtime solvable in general graphs, the graphs H ∈ {P₃,C₄} are the only connected ones that allow for a subexponentialtime algorithm in Ffree graphs for every F ∈ 𝒮 (unless the ETH fails).
BibTeX  Entry
@InProceedings{dvorak_et_al:LIPIcs.ISAAC.2022.30,
author = {Dvo\v{r}\'{a}k, Pavel and Masa\v{r}{\'\i}k, Tom\'{a}\v{s} and Novotn\'{a}, Jana and Krawczyk, Monika and Rz\k{a}\.{z}ewski, Pawe{\l} and \.{Z}uk, Aneta},
title = {{List Locally Surjective Homomorphisms in Hereditary Graph Classes}},
booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)},
pages = {30:130:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772587},
ISSN = {18688969},
year = {2022},
volume = {248},
editor = {Bae, Sang Won and Park, Heejin},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/17315},
URN = {urn:nbn:de:0030drops173154},
doi = {10.4230/LIPIcs.ISAAC.2022.30},
annote = {Keywords: Homomorphism, Hereditary graphs, Subexponentialtime algorithms}
}
Keywords: 

Homomorphism, Hereditary graphs, Subexponentialtime algorithms 
Collection: 

33rd International Symposium on Algorithms and Computation (ISAAC 2022) 
Issue Date: 

2022 
Date of publication: 

14.12.2022 