Superextensions and the depth of median graphs

Bandelt, Hans‐Jürgen; Vel, M. van de

doi:10.1016/0097-3165(91)90044-h

Cited by 43 publications

(44 citation statements)

References 11 publications

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“…The previous result generalizes a result of Bandelt and van de Vel [2] in the setting of median graphs stating that the collection of all balls in C has the Helly property. Note that d -balls are d -convex by Corollary 3.5.…”

Section: Problem 224supporting

confidence: 53%

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Packing subgroups in relatively hyperbolic groups

2009

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We introduce the bounded packing property for a subgroup of a countable discrete group G . This property gives a finite upper bound on the number of left cosets of the subgroup that are pairwise close in G . We establish basic properties of bounded packing and give many examples; for instance, every subgroup of a countable, virtually nilpotent group has bounded packing. We explain several natural connections between bounded packing and group actions on CAT.0/ cube complexes.Our main result establishes the bounded packing of relatively quasiconvex subgroups of a relatively hyperbolic group, under mild hypotheses. As an application, we prove that relatively quasiconvex subgroups have finite height and width, properties that strongly restrict the way families of distinct conjugates of the subgroup can intersect. We prove that an infinite, nonparabolic relatively quasiconvex subgroup of a relatively hyperbolic group has finite index in its commensurator. We also prove a virtual malnormality theorem for separable, relatively quasiconvex subgroups, which is new even in the word hyperbolic case. 20F65; 20F67, 20F69

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Section: Problem 224supporting

confidence: 53%

“…Maximizing over all such choices gives a uniform upper bound S on the diameter of the intersection I WD N R .gP /\N R .aH /. If a 0 H 0 is any other element of A, then (2) implies that N R .a 0 H 0 / intersects I .…”

Section: Bounded Packing In Relatively Hyperbolic Groupsmentioning

confidence: 99%

Packing subgroups in relatively hyperbolic groups

2009

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“…Median graphs and the related median structures have many nice characterizations and properties, investigated by several authors; [5,7,11,14,15,35,42,43,54,55,56] is a sample of papers on this subject.…”

Section: A Graph G Is Modular If and Only If It Is Triangle-free Anmentioning

confidence: 99%

Graphs of Some CAT(0) Complexes

Chepoi

2000

Advances in Applied Mathematics

214

251

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In this note, we characterize the graphs (1-skeletons) of some piecewise Euclidean simplicial and cubical complexes having nonpositive curvature in the sense of Gromov's CAT(0) inequality. Each such cell complex K is simply connected and obeys a certain flag condition. It turns out that if, in addition, all maximal cells are either regular Euclidean cubes or right Euclidean triangles glued in a special way, then the underlying graph G K is either a median graph or a hereditary modular graph without two forbidden induced subgraphs. We also characterize the simplicial complexes arising from bridged graphs, a class of graphs whose metric enjoys one of the basic properties of CAT (0) spaces. Additionally, we show that the graphs of all these complexes and some more general classes of graphs have geodesic combings and bicombings verifying the 1-or 2-fellow traveler property.

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“…However, using the following concept, the answer is clear. For a graph G, the simplex graph S(G) of G is the graph whose vertices are the complete subgraphs of G (including the empty graph), two vertices being adjacent if, as complete subgraphs of G, they differ in exactly one vertex; see [7]. It is easily seen that a simplex graph is a median graph, and hence a partial cube, by checking that it satisfies the definition of a median graph.…”

Section: Preliminaries Formentioning

confidence: 99%

Partial Cubes and Crossing Graphs

Klavžar¹,

Mulder²

2002

SIAM J. Discrete Math.

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Abstract. Partial cubes are defined as isometric subgraphs of hypercubes. For a partial cube G, its crossing graph G # is introduced as the graph whose vertices are the equivalence classes of the Djoković-Winkler relation Θ, two vertices being adjacent if they cross on a common cycle. It is shown that every graph is the crossing graph of some median graph and that a partial cube G is 2-connected if and only if G # is connected. A partial cube G has a triangle-free crossing graph if and only if G is a cube-free median graph. This result is used to characterize the partial cubes having a tree or a forest as its crossing graph. An expansion theorem is given for the partial cubes with complete crossing graphs. Cartesian products are also considered. In particular, it is proved that G # is a complete bipartite graph if and only if G is the Cartesian product of two trees.

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Superextensions and the depth of median graphs

Cited by 43 publications

References 11 publications

Packing subgroups in relatively hyperbolic groups

Packing subgroups in relatively hyperbolic groups

Graphs of Some CAT(0) Complexes

Partial Cubes and Crossing Graphs

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