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Structural Parameterizations of b-Coloring

Authors Lars Jaffke , Paloma T. Lima , Roohani Sharma

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Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • b-coloring
  • structural parameterization
  • XNLP
  • pathwidth
  • neighborhood diversity
  • twin cover

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Abstract

The b-Coloring problem, which given a graph G and an integer k asks whether G has a proper k-coloring such that each color class has a vertex adjacent to all color classes except its own, is known to be FPT parameterized by the vertex cover number and XP and 𝖶[1]-hard parameterized by clique-width. Its complexity when parameterized by the treewidth of the input graph remained an open problem. We settle this question by showing that b-Coloring is XNLP-complete when parameterized by the pathwidth of the input graph. Besides determining the precise parameterized complexity of this problem, this implies that b-Coloring parameterized by pathwidth is 𝖶[t]-hard for all t, and resolves the parameterized complexity of b-Coloring parameterized by treewidth. We complement this result by showing that b-Coloring is FPT when parameterized by neighborhood diversity and by twin cover, two parameters that generalize vertex cover to more dense graphs, but are incomparable to pathwidth.

Cite As Get BibTex

Lars Jaffke, Paloma T. Lima, and Roohani Sharma. Structural Parameterizations of b-Coloring. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 40:1-40:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ISAAC.2023.40

Author Details

Lars Jaffke
  • University of Bergen, Norway
Paloma T. Lima
  • IT University of Copenhagen, Denmark
Roohani Sharma
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany

Funding

  • Lima, Paloma T.: This work received funding from the Independent Research Fund Denmark grant agreement number 2098-00012B.

Acknowledgements

We would like to thank an anonymous reviewer for carefully reading our paper and for many useful comments.

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