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# Complexity of the List Homomorphism Problem in Hereditary Graph Classes

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## Cite As

Karolina Okrasa and Paweł Rzążewski. Complexity of the List Homomorphism Problem in Hereditary Graph Classes. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 54:1-54:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.STACS.2021.54

## Abstract

A homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H). For a fixed graph H, in the list homomorphism problem, denoted by LHom(H), we are given a graph G, whose every vertex v is equipped with a list L(v) ⊆ V(H). We ask if there exists a homomorphism f from G to H, in which f(v) ∈ L(v) for every v ∈ V(G). Feder, Hell, and Huang [JGT 2003] proved that LHom(H) is polynomial time-solvable if H is a so-called bi-arc-graph, and NP-complete otherwise. We are interested in the complexity of the LHom(H) problem in F-free graphs, i.e., graphs excluding a copy of some fixed graph F as an induced subgraph. It is known that if F is connected and is not a path nor a subdivided claw, then for every non-bi-arc graph the LHom(H) problem is NP-complete and cannot be solved in subexponential time, unless the ETH fails. We consider the remaining cases for connected graphs F. If F is a path, we exhibit a full dichotomy. We define a class called predacious graphs and show that if H is not predacious, then for every fixed t the LHom(H) problem can be solved in quasi-polynomial time in P_t-free graphs. On the other hand, if H is predacious, then there exists t, such that the existence of a subexponential-time algorithm for LHom(H) in P_t-free graphs would violate the ETH. If F is a subdivided claw, we show a full dichotomy in two important cases: for H being irreflexive (i.e., with no loops), and for H being reflexive (i.e., where every vertex has a loop). Unless the ETH fails, for irreflexive H the LHom(H) problem can be solved in subexponential time in graphs excluding a fixed subdivided claw if and only if H is non-predacious and triangle-free. On the other hand, if H is reflexive, then LHom(H) cannot be solved in subexponential time whenever H is not a bi-arc graph.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Graph coloring
• Theory of computation → Problems, reductions and completeness
• Theory of computation → Graph algorithms analysis
##### Keywords
• list homomorphism
• fine-grained complexity
• hereditary graph classes

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