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arXiv · 2605.03965

Tree-independence number of $P_5$-free graphs with no large bicliques

Václav Blažej·J. Pascal Gollin·Tomáš Hons·Tomáš Masařík·Martin Milanič·Paweł Rzążewski·Ondřej Suchý·Alexandra Wesolek
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Abstract

The tree-independence number of a graph is the minimum, over all tree-decompositions of the graph, of the maximum size of an independent set contained in a bag. Graph classes of bounded tree-independence number have strong structural and algorithmic properties, but the parameter can be unbounded even in quite restricted classes. In particular, the presence of an induced biclique $K_{\ell,\ell}$ forces tree-independence number at least $\ell$. This leads to the question whether large induced bicliques are the only obstruction to bounded tree-independence number in natural hereditary classes. A conjecture of Dallard, Krnc, Kwon, Milani\v{c}, Munaro, \v{S}torgel, and Wiederrecht states that for all positive integers $t$ and $\ell$, every $\{P_t,K_{\ell,\ell}\}$-free graph has bounded tree-independence number. We prove this conjecture for $t=5$ by showing that every $\{P_5,K_{\ell,\ell}\}$-free graph has tree-independence number at most $4\ell$. We also obtain related bounds for the weaker parameter of $\alpha$-degeneracy.

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Václav Blažej, J. Pascal Gollin, Tomáš Hons, Tomáš Masařík, Martin Milanič, Paweł Rzążewski, Ondřej Suchý, Alexandra Wesolek. 2026-05-05. Tree-independence number of $P_5$-free graphs with no large bicliques. https://arxiv.org/abs/2605.03965

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