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Treewidth Versus Clique Number in Graph Classes with a Forbidden Structure

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Graph-Theoretic Concepts in Computer Science (WG 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12301))

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Abstract

Treewidth is an important graph invariant, relevant for both structural and algorithmic reasons. A necessary condition for a graph class to have bounded treewidth is the absence of large cliques. We study graph classes in which this condition is also sufficient, which we call \((\text {tw},\omega )\)-bounded. Such graph classes are known to have useful algorithmic applications related to variants of the clique and k-coloring problems. We consider six well-known graph containment relations: the minor, topological minor, subgraph, induced minor, induced topological minor, and induced subgraph relations. For each of them, we give a complete characterization of the graphs H for which the class of graphs excluding H is \((\text {tw},\omega )\)-bounded. Our results imply that the class of 1-perfectly orientable graphs is \((\text {tw},\omega )\)-bounded, answering a question of Brešar, Hartinger, Kos, and Milanič from 2018. We also reveal some further algorithmic implications of \((\text {tw},\omega )\)-boundedness related to list k-coloring and clique problems.

This research was funded in part by the Slovenian Research Agency (I0-0035, research program P1-0285, research projects J1-9110, N1-0102, and N1-0160, and a Young Researchers Grant).

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Authors and Affiliations

  1. FAMNIT, University of Primorska, Koper, Slovenia

    Clément Dallard, Martin Milanič & Kenny Štorgel

  2. IAM, University of Primorska, Koper, Slovenia

    Clément Dallard & Martin Milanič

  3. Faculty of Information Studies, Novo mesto, Slovenia

    Kenny Štorgel

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  1. Clément Dallard
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  2. Martin Milanič
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Correspondence to Martin Milanič .

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  1. University of Leeds, Leeds, UK

    Isolde Adler

  2. University of Leeds, Leeds, UK

    Haiko Müller

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Dallard, C., Milanič, M., Štorgel, K. (2020). Treewidth Versus Clique Number in Graph Classes with a Forbidden Structure. In: Adler, I., Müller, H. (eds) Graph-Theoretic Concepts in Computer Science. WG 2020. Lecture Notes in Computer Science(), vol 12301. Springer, Cham. https://doi.org/10.1007/978-3-030-60440-0_8

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  • DOI: https://doi.org/10.1007/978-3-030-60440-0_8

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