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.
Davies,String graphs are quasi-isometric to planar graphs
4 Pith papers cite this work. Polarity classification is still indexing.
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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.
Finitely presented groups with k-planar Cayley graphs have finite-index subgroups with planar Cayley graphs; k-planar coarsely simply connected quasi-transitive graphs are quasi-isometric to planar graphs.
Sparse string graphs on fixed surfaces and sparse region intersection graphs over proper minor-closed classes have linear expansion with bounds within a constant factor of optimal.
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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Almost planar finitely presented groups
Finitely presented groups with k-planar Cayley graphs have finite-index subgroups with planar Cayley graphs; k-planar coarsely simply connected quasi-transitive graphs are quasi-isometric to planar graphs.
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Sparse String Graphs and Region Intersection Graphs over Minor-Closed Classes have Linear Expansion
Sparse string graphs on fixed surfaces and sparse region intersection graphs over proper minor-closed classes have linear expansion with bounds within a constant factor of optimal.