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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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.
Graphs excluding any fixed H as a d-fat minor admit balanced separators coverable by O(n^{1/2+ε}) radius-r balls, with a poly-time algorithm to find the separator or the fat model.
citing papers explorer
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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.
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Coarse Balanced Separators in Fat-Minor-Free Graphs
Graphs excluding any fixed H as a d-fat minor admit balanced separators coverable by O(n^{1/2+ε}) radius-r balls, with a poly-time algorithm to find the separator or the fat model.