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$\chi$-Boundedness and Neighbourhood Complexity of Bounded Merge-Width Graphs

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arxiv 2504.08266 v2 pith:RC2DYNKZ submitted 2025-04-11 math.CO cs.DM

Marthe Bonamy , Colin Geniet This is my paper · ORCID
classification math.COcs.DM
keywords boundedgraphsmerge-widthcomplexityexpansionneighbourhoodtheytwin-width
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abstract

Merge-width, recently introduced by Dreier and Toru\'nczyk, is a common generalisation of bounded expansion classes and twin-width for which the first-order model checking problem remains tractable. We prove that a number of basic properties shared by bounded expansion and bounded twin-width graphs also hold for bounded merge-width graphs: they are $\chi$-bounded, they satisfy the strong Erd\H{o}s-Hajnal property, and their neighbourhood complexity is linear.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Neighbourhood complexity and identification problems for graphs of bounded treewidth and pathwidth

    cs.DM 2026-07 conditional novelty 8.0 of 10

    For graphs of treewidth w and pathwidth w, neighbourhood complexity is exactly (k-w+1)2^w + w and (k-w+2)2^(w-1)+2k-w-2, with matching constructions; forests give floor(7k/3).

  2. Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes

    cs.DM 2026-07 accept novelty 7.5 of 10

    Every monadically dependent hereditary graph class has almost-linear neighborhood complexity and n^{o(1)} radius-1 merge-width, witnessed by an efficient construction-sequence algorithm.

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