Bounded tree-independence for {P_r, K_{t,t}}-free graphs (Dallard et al. conjecture)

Establish whether, for every pair of positive integers r and t, the class of graphs that contain no induced subgraph isomorphic to the r-vertex path P_r and no induced subgraph isomorphic to the complete bipartite graph K_{t,t} has bounded tree-independence number.

Background

Tree-independence number generalizes treewidth and admits algorithmic metatheorems, making its boundedness in hereditary classes a central question. Dallard et al. proposed that excluding both a long induced path and a large complete bipartite graph might suffice to bound this parameter.

The present paper proves the conjecture for r ≤ 6 (for all t), making partial progress, but the general case remains unresolved.

References

While the conjecture was recently disproved by Chudnovsky and Trotignon (see), it is still open for classes defined by finitely many forbidden induced subgraphs, even in the following special case. For any two positive integers~$r$ and~$t$, the class of~${{P_r, K_{t,t}}$-free graphs has bounded tree-independence number.

— Tree-independence number and forbidden induced subgraphs: excluding a $6$-vertex path and a $(2,t)$-biclique  (2604.01999 - Chudnovsky et al., 2 Apr 2026) in Conjecture 1, Section 1 (Introduction)