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

arxiv(2024)

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摘要
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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