Tree-independence number and forbidden induced subgraphs: excluding a $6$-vertex path and a $(2,t)$-biclique

TL;DR

This paper proves that graphs excluding certain small induced subgraphs have bounded tree-independence number, advancing understanding of graph structure related to forbidden subgraphs.

Contribution

It establishes a bound on the tree-independence number for graphs excluding a 6-vertex path and a (2,t)-biclique, partially addressing a conjecture.

Findings

Graphs with no induced 6-vertex path or (2,t)-biclique have bounded tree-independence number.

The result applies for every t ≥ 2, with a specific bound s depending on t.

Progress on a conjecture by Dallard et al. regarding graph structure.

Abstract

We show that for every positive integer ${t \geq 2}$ there exists an integer $s$ such that every graph that contains no induced subgraph isomorphic to either the $6$-vertex path or the $(2,t)$-biclique, the complete bipartite graph $K_{2,t}$, has tree-independence number at most $s$. This result makes partial progress on a conjecture of Dallard, Krnc, Kwon, Milani\v{c}, Munaro, \v{S}torgel, and Wiederrecht.