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arXiv · 2305.16258

Tree independence number I. (Even hole, diamond, pyramid)-free graphs

Tara Abrishami·Bogdan Alecu·Maria Chudnovsky·Sepehr Hajebi·Sophie Spirkl·Kristina Vušković
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Abstract

The tree-independence number tree-$\alpha$, first defined and studied by Dallard, Milani\v{c} and \v{S}torgel, is a variant of treewidth tailored to solving the maximum independent set problem. Over a series of papers, Abrishami et al. developed the so-called central bag method to study induced obstructions to bounded treewidth. Among others, they showed that, in a certain superclass $\mathcal C$ of (even hole, diamond, pyramid)-free graphs, treewidth is bounded by a function of the clique number. In this paper, we relax the bounded clique number assumption, and show that $\mathcal C$ has bounded tree-$\alpha$. Via existing results, this yields a polynomial time algorithm for the maximum independent set problem in this class. Our result also corroborates, for this class of graphs, a conjecture of Dallard, Milani\v{c} and \v{S}torgel that in a hereditary graph class, tree-$\alpha$ is bounded if and only if the treewidth is bounded by a function of the clique number.

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Tara Abrishami, Bogdan Alecu, Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl, Kristina Vušković. 2023-05-25. Tree independence number I. (Even hole, diamond, pyramid)-free graphs. https://doi.org/10.1002/jgt.23104

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