eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-03-10
54:1
54:17
10.4230/LIPIcs.STACS.2021.54
article
Complexity of the List Homomorphism Problem in Hereditary Graph Classes
Okrasa, Karolina
1
2
https://orcid.org/0000-0003-1414-3507
Rzążewski, Paweł
1
2
https://orcid.org/0000-0001-7696-3848
Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland
Institute of Informatics, University of Warsaw, Poland
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol187-stacs2021/LIPIcs.STACS.2021.54/LIPIcs.STACS.2021.54.pdf
list homomorphism
fine-grained complexity
hereditary graph classes