Sim-Width, Induced Matching Treewidth, and Tree-Independence Number in Induced $K_{t,t}$-Free Graphs

RESUMO

The tree-independence number $tree\text{-}\alpha(G)$, the induced matching treewidth $tree\text{-}\mu(G)$, and the sim-width $simw(G)$ are graph parameters defined in terms of tree or branch decompositions. We establish two polynomial bounds for the tree-independence number of induced $K_{t,t}$-free graphs, one in terms of sim-width and the other in terms of induced matching treewidth. Abrishami et al. (SIDMA, 2025) and Brettell et al. (EJC, 2025) asked whether bounded sim-width, together with the exclusion of an induced $K_{t,t}$, implies bounded tree-independence number. We answer this question by proving that, for integers $t\geq 2$ and $s\geq 1$, every induced $K_{t,t}$-free graph $G$ with $simw(G)\leq s$ satisfies $tree\text{-}\alpha(G)=O_t\left((s+1)^{2t^2-2t}\right)$. This also proves a polynomial strengthening of a conjecture of Bešter Štorgel et al. (arXiv, 2026) concerning induced $K_{1,t}$-free graphs and improves a theorem of Alon et al. (arXiv, 2025) by reducing the exponent from $3t^2+1$ to $2t^2-2t$. Alon et al. (arXiv, 2025) asked whether, for fixed induced matching treewidth, the tree-independence number is polynomially bounded in $t$. Using a VC-dimension argument, we answer this question affirmatively by showing that, for integers $\mu\geq 1$ and $t\geq 2$, every induced $K_{t,t}$-free graph $G$ with $tree\text{-}\mu(G)\leq\mu$ satisfies $tree\text{-}\alpha(G)=t^{O_\mu(1)}$.