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DOI: 10.4230/LIPIcs.ESA.2020.35
Finding Large H-Colorable Subgraphs in Hereditary Graph Classes

Authors: Maria Chudnovsky, Jason King, Michał Pilipczuk, Paweł Rzążewski, and Sophie Spirkl

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Abstract
We study the Max Partial H-Coloring problem: given a graph G, find the largest induced subgraph of G that admits a homomorphism into H, where H is a fixed pattern graph without loops. Note that when H is a complete graph on k vertices, the problem reduces to finding the largest induced k-colorable subgraph, which for k = 2 is equivalent (by complementation) to Odd Cycle Transversal. We prove that for every fixed pattern graph H without loops, Max Partial H-Coloring can be solved: - in {P₅,F}-free graphs in polynomial time, whenever F is a threshold graph; - in {P₅,bull}-free graphs in polynomial time; - in P₅-free graphs in time n^𝒪(ω(G)); - in {P₆,1-subdivided claw}-free graphs in time n^𝒪(ω(G)³). Here, n is the number of vertices of the input graph G and ω(G) is the maximum size of a clique in G. Furthermore, by combining the mentioned algorithms for P₅-free and for {P₆,1-subdivided claw}-free graphs with a simple branching procedure, we obtain subexponential-time algorithms for Max Partial H-Coloring in these classes of graphs. Finally, we show that even a restricted variant of Max Partial H-Coloring is NP-hard in the considered subclasses of P₅-free graphs, if we allow loops on H.
Cite as

Maria Chudnovsky, Jason King, Michał Pilipczuk, Paweł Rzążewski, and Sophie Spirkl. Finding Large H-Colorable Subgraphs in Hereditary Graph Classes. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 35:1-35:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chudnovsky_et_al:LIPIcs.ESA.2020.35,
  author =	{Chudnovsky, Maria and King, Jason and Pilipczuk, Micha{\l} and Rz\k{a}\.{z}ewski, Pawe{\l} and Spirkl, Sophie},
  title =	{{Finding Large H-Colorable Subgraphs in Hereditary Graph Classes}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{35:1--35:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.35},
  URN =		{urn:nbn:de:0030-drops-129019},
  doi =		{10.4230/LIPIcs.ESA.2020.35},
  annote =	{Keywords: homomorphisms, hereditary graph classes, odd cycle transversal}
}
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