Graphs with cycle spaces generated by bounded-length cycles have the coarse Menger property, with corollaries for hyperbolic graphs, finitely presented groups, and planar graphs with bounded faces.
Asymptotic structure. VI. Distant paths across a disc
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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.
Locally finite graphs with an excluded finite minor have the weak coarse Menger property with f depending only on k and g linear in r independent of k.
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A coarse Menger theorem for hyperbolic graphs, finitely presented groups, and more
Graphs with cycle spaces generated by bounded-length cycles have the coarse Menger property, with corollaries for hyperbolic graphs, finitely presented groups, and planar graphs with bounded faces.
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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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Coarse Menger property of quasi-minor excluded graphs and length spaces
Locally finite graphs with an excluded finite minor have the weak coarse Menger property with f depending only on k and g linear in r independent of k.