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Finding Large H-Colorable Subgraphs in Hereditary Graph Classes

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

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ACM Subject Classification
  • Mathematics of computing → Graph coloring
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • homomorphisms
  • hereditary graph classes
  • odd cycle transversal

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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 Get BibTex

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) https://doi.org/10.4230/LIPIcs.ESA.2020.35

Author Details

Maria Chudnovsky
  • Princeton University, NJ, USA
Jason King
  • Princeton University, NJ, USA
Michał Pilipczuk
  • Institute of Informatics, University of Warsaw, Poland
Paweł Rzążewski
  • Warsaw University of Technology, Faculty of Mathematics and Information Science, Poland
  • University of Warsaw, Institute of Informatics, Poland
Sophie Spirkl
  • Princeton University, NJ, USA

Funding

  • Chudnovsky, Maria: This material is based upon work supported in part by the U. S. Army Research Office under grant number W911NF-16-1-0404, and by NSF grant DMS-1763817.
  • Pilipczuk, Michał: This work is a part of project TOTAL that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 677651).
  • Rzążewski, Paweł: Supported by Polish National Science Centre grant no. 2018/31/D/ST6/00062.
  • Spirkl, Sophie: This material is based upon work supported by the National Science Foundation under Award No. DMS1802201.

Acknowledgements

We acknowledge the welcoming and productive atmosphere at Dagstuhl Seminar 19271 "Graph Colouring: from Structure to Algorithms", where this work has been initiated.

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