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Grid Minors and Products

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arxiv 2402.14181 v3 pith:YMBRQBEU submitted 2024-02-21 math.CO cs.DM

Vida Dujmović , Pat Morin , David R. Wood , David Worley This is my paper · ORCID
classification math.COcs.DM
keywords gridomegaproductsqrtcartesiangraphsminorminors
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abstract

Motivated by recent developments regarding the product structure of planar graphs, we study relationships between treewidth, grid minors, and graph products. We show that the Cartesian product of any two connected $n$-vertex graphs contains an $\Omega(\sqrt{n})\times\Omega(\sqrt{n})$ grid minor. This result is tight: The lexicographic product (which includes the Cartesian product as a subgraph) of a star and any $n$-vertex tree has no $\omega(\sqrt{n})\times\omega(\sqrt{n})$ grid minor.

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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. Treewidth of Products of Graphs with High Treewidth

    math.CO 2026-07 conditional novelty 8.0 of 10

    The treewidth of the strong product of two graphs is at least the product of their treewidth-plus-ones, minus one; analogous bounds hold for pathwidth and Cartesian products.

  2. Short Paths in the Planar Graph Product Structure Theorem

    math.CO 2025-02 conditional novelty 8.0 of 10

    Every n-vertex planar graph is contained in H ⊠ P ⊠ K_c for some planar H of treewidth 3 and a path P of length O((tw(G)+1)^(1-ε) n^ε).

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