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

2026-05-05

AI Summary

This work investigates the boundedness of tree-independence number in graph classes excluding the induced path $P_5$ and large complete bipartite graphs $K_{\ell,\ell}$. By integrating graph decomposition techniques, forbidden induced subgraph structural theory, and extremal combinatorial methods, the authors establish for the first time that every $\{P_5, K_{\ell,\ell}\}$-free graph has tree-independence number at most $4\ell$, along with a corresponding upper bound on its $\alpha$-degeneracy. This result confirms a conjecture by Dallard et al. regarding the bounded tree-independence number in $P_5$-free graphs and demonstrates that large induced bicliques are the sole obstruction to boundedness. The findings provide a foundational theoretical basis for algorithmic design and structural analysis within these graph classes.

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, 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.

Problem

• tree-independence number
• P5-free graphs
• bicliques
• hereditary graph classes
• induced subgraphs

Innovation

• tree-independence number
• P5-free graphs
• bicliques
• hereditary graph classes
• alpha-degeneracy