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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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.
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.
Every connected graph has a tree-decomposition with bounded-diameter adhesion sets and coarsely inseparable bags, yielding a metric analogue of the block-cutvertex tree.
citing papers explorer
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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 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.
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A coarse block-cutvertex tree-decomposition
Every connected graph has a tree-decomposition with bounded-diameter adhesion sets and coarsely inseparable bags, yielding a metric analogue of the block-cutvertex tree.