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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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.
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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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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.