Induced matching treewidth and tree-independence number, revisited

TL;DR

This work analyzes two width-like graph parameters defined via tree decompositions: the tree-independence number $\mathop{\mathrm{tree\text{-}\alpha}}(G)$ and the induced matching treewidth $\mathop{\mathrm{tree\text{-}\mu}}(G)$. It shows that while $\mathop{\mathrm{tree\text{-}\mu}}(G)\le\mathop{\mathrm{tree\text{-}\alpha}}(G)$ in general and complete bipartite graphs separate the two, a polynomial relationship between the parameters holds for $K_{t,t}$-free graph classes, improving earlier exponential bounds. The core strategy combines the Kövári–Sós–Turán theorem with two Ramsey-type lemmas to control induced matchings and independent sets within tree decompositions, yielding $\mathop{\mathrm{tree\text{-}\alpha}}(G)=\mathcal{O}_t(\mu^{3t^2+1})$ when $\mathop{\mathrm{tree\text{-}\mu}}(G)\le\mu$. The paper also provides a probabilistic construction showing that no polynomial bound in $t$ alone can hold in general, and outlines open questions about joint bounds, induced biclique numbers, and $\,\chi$-boundedness.

Abstract

We study two graph parameters defined via tree decompositions: tree-independence number and induced matching treewidth. Both parameters are defined similarly as treewidth, but with respect to different measures of a tree decomposition $\mathcal{T}$ of a graph $G$: for tree-independence number, the measure is the maximum size of an independent set in $G$ included in some bag of $\mathcal{T}$, while for the induced matching treewidth, the measure is the maximum size of an induced matching in $G$ such that some bag of $\mathcal{T}$ contains at least one endpoint of every edge of the matching. While the induced matching treewidth of any graph is bounded from above by its tree-independence number, the family of complete bipartite graphs shows that small induced matching treewidth does not imply small tree-independence number. On the other hand, Abrishami, Briański, Czyżewska, McCarty, Milanič, Rzążewski, and Walczak~[SIAM Journal on Discrete Mathematics, 2025] showed that, if a fixed biclique $K_{t,t}$ is excluded as an induced subgraph, then the tree-independence number is bounded from above by some function of the induced matching treewidth. The function resulting from their proof is exponential even for fixed $t$, as it relies on multiple applications of Ramsey's theorem. In this note we show, using the Kövári-Sós-Turán theorem, that for any class of $K_{t,t}$-free graphs, the two parameters are in fact polynomially related.

Theorems & Definitions (9)

• Theorem 1.1: Abrishami et al.
• Theorem 1.2
• Theorem 3.1: Kövári-Sós-Turán
• Corollary 3.2
• Theorem 3.3: Turán
• Lemma 3.4
• Lemma 3.5
• Theorem 4.1
• Lemma 5.1