Bounded tree-independence for {P_r, K_{2,t}}-free graphs

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_{2,t} has bounded tree-independence number.

Background

This conjecture is a specialization of the {P_r, K_{t,t}} conjecture that replaces K_{t,t} by K_{2,t}. It is highlighted as an interesting open case by earlier work.

The present paper confirms the conjecture for r ≤ 6 and all t, leaving the higher r cases open.

References

Furthermore, they observed that for fixed~$r$ and~$t$, tree-independence number is bounded in any class of ${{P_r,K_{1,t}}$\nobreakdash-free graphs (see also) and suggested the following as an interesting open case. For any two positive integers~$r$ and~$t$, the class of ${{P_r, K_{2,t}}$-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 2, Section 1 (Introduction)