Excluding a clique or a biclique in graphs of bounded induced matching treewidth
arxiv(2024)
Abstract
When 𝒯 is a tree decomposition of a graph G, we write
μ(𝒯) for the maximum size of an induced matching in G all of
whose edges intersect one bag of 𝒯. The induced matching treewidth
of a graph G is the minimum value of μ(𝒯) over all tree
decompositions 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 structural 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 χ-bounded. Our results confirm two
conjectures from a recent manuscript of Lima et al. [arXiv 2024].
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