Smallest open cases: {P_7, K_{2,2}}-free and {P_5, K_{3,3}}-free graphs

Determine whether the tree-independence number is bounded for (i) the class of graphs excluding P_7 and K_{2,2} as induced subgraphs and (ii) the class of graphs excluding P_5 and K_{3,3} as induced subgraphs.

Background

These two instances are identified as the minimal unresolved cases following the paper’s main result that settles {P_r, K_{2,t}} for r ≤ 6.

Resolving either case would advance understanding of the {P_r, K_{t,t}} and {P_r, K_{2,t}} conjectures in specific low-parameter regimes.

References

In particular, the smallest open cases involve determining (un)boundedness of tree-independence number in the classes of ${P_7,K_{2,2}}$-free graphs and ${P_5,K_{3,3}}$-free graphs.

— 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 Section 1 (Introduction)