Excluding a clique or a biclique in graphs of bounded induced matching treewidth

Abstract

For a tree decomposition $\mathcal{T}$ of a graph $G$, let $\mu(\mathcal{T})$ denote the maximum size of an induced matching in $G$ with the property that some bag of $\mathcal{T}$ contains at least one endpoint of every edge of the matching. The induced matching treewidth of a graph $G$ is the minimum value of $\mu(\mathcal{T})$ over all tree decompositions $\mathcal{T}$ of $G$. Classes of graphs with bounded induced matching treewidth admit polynomial-time algorithms for a number of problems, including INDEPENDENT SET, $k$-COLORING, ODD CYCLE TRANSVERSAL, and FEEDBACK VERTEX SET. In this paper, we focus on combinatorial properties of such classes. First, we show that graphs with bounded induced matching treewidth that exclude a fixed biclique as an induced subgraph have bounded tree-independence number, which is another well-studied parameter defined in terms of tree decompositions. This sufficient condition about excluding a biclique is also necessary, as bicliques have unbounded tree-independence number. Second, we show that graphs with bounded induced matching treewidth that exclude a fixed clique have bounded chromatic number, that is, classes of graphs with bounded induced matching treewidth are $\chi$-bounded. The two results confirm two conjectures due to Lima et al. [ESA 2024].