{"ID":22938767,"CreatedAt":"2026-09-17T01:37:05.451791504Z","UpdatedAt":"2026-09-17T01:37:05.451791504Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.18648","arxiv_id":"2609.18648","title":"Sim-Width, Induced Matching Treewidth, and Tree-Independence Number in Induced $K_{t,t}$-Free Graphs","abstract":"The tree-independence number $tree\\text{-}α(G)$, the induced matching treewidth $tree\\text{-}μ(G)$, 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 $tree\\text{-}α(G)=O_t\\left((s+1)^{2t^2-2t}\\right)$. 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 $μ\\geq 1$ and $t\\geq 2$, every induced $K_{t,t}$-free graph $G$ with $tree\\text{-}μ(G)\\leqμ$ satisfies $tree\\text{-}α(G)=t^{O_μ(1)}$.","short_abstract":"The tree-independence number $tree\\text{-}α(G)$, the induced matching treewidth $tree\\text{-}μ(G)$, 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 si...","url_abs":"https://arxiv.org/abs/2609.18648","url_pdf":"https://arxiv.org/pdf/2609.18648v1","authors":"[\"Mengyuan Niu\",\"Xiumei Wang\"]","published":"2026-09-16T13:32:32Z","proceeding":"math.CO","tasks":"[\"math.CO\"]","methods":"[]","has_code":false}