Tree-independence number of $P_5$-free graphs with no large bicliques
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| Main Authors: | , , , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910192220766208 |
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| author | Blažej, Václav Gollin, J. Pascal Hons, Tomáš Masařík, Tomáš Milanič, Martin Rzążewski, Paweł Suchý, Ondřej Wesolek, Alexandra |
| author_facet | Blažej, Václav Gollin, J. Pascal Hons, Tomáš Masařík, Tomáš Milanič, Martin Rzążewski, Paweł Suchý, Ondřej Wesolek, Alexandra |
| contents | 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, Milanič, Munaro, Štorgel, 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 $α$-degeneracy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03965 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Tree-independence number of $P_5$-free graphs with no large bicliques Blažej, Václav Gollin, J. Pascal Hons, Tomáš Masařík, Tomáš Milanič, Martin Rzążewski, Paweł Suchý, Ondřej Wesolek, Alexandra Combinatorics Discrete Mathematics 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, Milanič, Munaro, Štorgel, 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 $α$-degeneracy. |
| title | Tree-independence number of $P_5$-free graphs with no large bicliques |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2605.03965 |