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

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; however, 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 $\ell$, $\{P_t,K_{\ell ,\ell }\}$-free graphs have 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 $4\ell -4$. We also obtain related bounds for the weaker parameter of $\alpha$-degeneracy and answer a question of Hilaire, Milanič, and Vasić whether tree-independence number of $\{P_5,K_{\ell ,\ell }\}$-free graphs exceeds $\ell$ by at most an additive constant.

An abridged version of this manuscript was published at the European Symposium on Algorithms (ESA 2026)