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.
Hickingbotham
5 Pith papers cite this work. Polarity classification is still indexing.
5
Pith papers citing it
abstract
In this paper, we characterise graphs that are quasi-isometric to graphs with bounded treewidth. Specifically, we prove that a graph is quasi-isometric to a graph with bounded treewidth if and only if it has a tree-decomposition where each bag consists of a bounded number of balls of bounded diameter. This result extends a characterisation by Berger and Seymour (2024) of graphs that are quasi-isometric to trees. Additionally, we characterise graphs that are quasi-isometric to graphs with bounded pathwidth and graphs that are quasi-isometric to graphs with bounded linewidth. As an application of these results, we show that graphs with bounded rank-width, graphs with bounded tree independence number, and graphs with bounded sim-width are quasi-isometric to graphs with bounded treewidth.
citation-role summary
background 2
citation-polarity summary
fields
math.CO 5years
2026 5roles
background 2polarities
background 2representative citing papers
Verifies stronger coarse balanced separator conjecture for all r in K_{t,t}-induced-minor-free graphs of bounded clique number via a polynomial-size hitting set Z for large balls on any Y.
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.
Every graph admits a tree decomposition with small-diameter adhesion sets where same-bag vertices cannot be separated by small, distant vertex sets.
citing papers explorer
-
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.
-
Coarse Balanced Separators in Biclique-Induced-Minor-Free Graphs
Verifies stronger coarse balanced separator conjecture for all r in K_{t,t}-induced-minor-free graphs of bounded clique number via a polynomial-size hitting set Z for large balls on any Y.
-
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.
-
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.
-
A coarse block-cut tree theorem
Every graph admits a tree decomposition with small-diameter adhesion sets where same-bag vertices cannot be separated by small, distant vertex sets.