Induced matching treewidth and tree-independence number, revisited

TL;DR

This paper investigates the relationship between two graph parameters, tree-independence number and induced matching treewidth, showing they are polynomially related in $K_{t,t}$-free graphs, improving previous exponential bounds.

Contribution

The paper proves that in $K_{t,t}$-free graphs, the tree-independence number and induced matching treewidth are polynomially related, refining earlier exponential bounds.

Findings

Polynomial relation between parameters in $K_{t,t}$-free graphs

Improves upon previous exponential bounds

Uses Kővári-Sós-Turán theorem for proof

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, Bria\'nski, Czy\.zewska, McCarty, Milani\v{c}, Rz\k{a}\.zewski, 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 K\"ov\'ari-S\'os-Tur\'an theorem, that for any class of $K_{t,t}$-free graphs, the two parameters are in fact polynomially related.