Edge separators for graphs excluding a minor

12/21/2022

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by   Gwenaël Joret, et al.

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We prove that every n-vertex K_t-minor-free graph G of maximum degree Δ has a set F of O(t^2(log t)^1/4√(Δ n)) edges such that every component of G - F has at most n/2 vertices. This is best possible up to the dependency on t and extends earlier results of Diks, Djidjev, Sykora, and Vrťo (1993) for planar graphs, and of Sykora and Vrťo (1993) for bounded-genus graphs. Our result is a consequence of the following more general result: The line graph of G is isomorphic to a subgraph of the strong product H ⊠ K_⌊ p ⌋ for some graph H with treewidth at most t-2 and p = √((t-3)Δ |E(G)|) + Δ.