In planar and bounded-genus graphs, absence of k pairwise d-far S-T paths implies a vertex set of size f(d,k) whose d-neighborhood intersects every S-T path.
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Complete Riemannian surfaces quasi-isometric to bounded-degree graphs admit triangulations with quasi-isometric 1-skeletons, and the authors identify the precise condition for an if-and-only-if statement.
Every graph admits a tree decomposition with small-diameter adhesion sets where same-bag vertices cannot be separated by small, distant vertex sets.
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A coarse Menger's Theorem for planar and bounded genus graphs
In planar and bounded-genus graphs, absence of k pairwise d-far S-T paths implies a vertex set of size f(d,k) whose d-neighborhood intersects every S-T path.
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Triangulating surfaces quasi-isometrically
Complete Riemannian surfaces quasi-isometric to bounded-degree graphs admit triangulations with quasi-isometric 1-skeletons, and the authors identify the precise condition for an if-and-only-if statement.
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A coarse block-cut tree theorem
Every graph admits a tree decomposition with small-diameter adhesion sets where same-bag vertices cannot be separated by small, distant vertex sets.