Median decompositions arise from any system of vertex cuts via Sageev's dual median graph, are uniquely minimal, satisfy median-width equals clique number on all graphs, and characterize proper geometric actions of groups on median graphs through canonical decompositions of Cayley graphs.
Distel,An alternative characterisation of graphs quasi-isometric to graphs of bounded treewidth
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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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Vertex cuts and median decompositions
Median decompositions arise from any system of vertex cuts via Sageev's dual median graph, are uniquely minimal, satisfy median-width equals clique number on all graphs, and characterize proper geometric actions of groups on median graphs through canonical decompositions of Cayley graphs.
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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.