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Maximum List r-Colorable Induced Subgraphs in kP₃-Free Graphs

Authors Esther Galby , Paloma T. Lima , Andrea Munaro , Amir Nikabadi

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ACM Subject Classification
  • Mathematics of computing → Graph coloring
  • Theory of computation → Graph algorithms analysis
Keywords
  • Hereditary classes
  • list coloring
  • odd cycle transversal
  • independent set

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Abstract

We show that, for every fixed positive integers r and k, Max-Weight List r-Colorable Induced Subgraph admits a polynomial-time algorithm on kP₃-free graphs. This problem is a common generalization of Max-Weight Independent Set, Odd Cycle Transversal and List r-Coloring, among others. Our result has several consequences. 
First, it implies that, for every fixed r ≥ 5, assuming 𝖯 ≠ NP, Max-Weight List r-Colorable Induced Subgraph is polynomial-time solvable on H-free graphs if and only if H is an induced subgraph of either kP₃ or P₅+kP₁, for some k ≥ 1. Second, it makes considerable progress toward a complexity dichotomy for Odd Cycle Transversal on H-free graphs, allowing to answer a question of Agrawal, Lima, Lokshtanov, Rzążewski, Saurabh, and Sharma [ACM Trans. Algorithms 2025]. Third, it gives a short and self-contained proof of the known result of Chudnovsky, Hajebi, and Spirkl [Combinatorica 2024] that List r-Coloring on kP₃-free graphs is polynomial-time solvable for every fixed r and k.
We also consider two natural distance-d generalizations of Max-Weight Independent Set and List r-Coloring and provide polynomial-time algorithms on kP₃-free graphs for every fixed integers r, k, and d ≥ 6.

Cite As Get BibTex

Esther Galby, Paloma T. Lima, Andrea Munaro, and Amir Nikabadi. Maximum List r-Colorable Induced Subgraphs in kP₃-Free Graphs. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 40:1-40:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.40

Author Details

Esther Galby
  • Chalmers University of Technology, Gothenburg, Sweden
  • University of Gothenburg, Sweden
Paloma T. Lima
  • IT University of Copenhagen, Denmark
Andrea Munaro
  • Department of Mathematical, Physical and Computer Sciences, University of Parma, Italy
Amir Nikabadi
  • IT University of Copenhagen, Denmark

Funding

  • Lima, Paloma T.: Supported by Independent Research Fund Denmark grant agreement number 2098-00012B.
  • Nikabadi, Amir: Supported by Independent Research Fund Denmark grant agreement number 2098-00012B.

References

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