Matching binary convexities

Vel, M. van de

doi:10.1016/0166-8641(83)90019-6

Cited by 32 publications

(24 citation statements)

References 8 publications

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“…Theorem 6.1 essentially facilitates this. Moreover, most of such properties have been already known for median graphs; [11,14,15,27,42,43,54,55] are just a few sources. On the other hand, the CAT(0) property sheds a new light on median complexes.…”

Section: Properties Of Median Complexesmentioning

confidence: 98%

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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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Section: Properties Of Median Complexesmentioning

confidence: 98%

“…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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“…One of the fundamental geometric results on median spaces is the following (reformulated) "Amalgamation Theorem". Early versions of this result were obtained by Isbell [7] and the author [14].…”

Section: Median Convexity In Cubical Polyhedramentioning

confidence: 93%

“…Consequently, p is an isomorphism between the median spaces H, p (H). On the other hand [14], the subspace H ∪ p (H) of G is convex and its relative convexity corresponds with the product convexity of p (H) × {0, 1} (where the second factor is given the "discrete" convexity). It follows easily that there is an isomorphism…”

Section: Theoremmentioning

confidence: 99%

Collapsible polyhedra and median spaces

Vel

1998

Proc. Amer. Math. Soc.

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Abstract. It is shown that a collapsible, compact, connected, simplicial polyhedron admits a cubical subdivision and a median convexity, such that all cubes are convex subspaces with a convexity of subcubes. Conversely, a compact, connected, cubical polyhedron with a convexity as described admits a collapsible simplicial subdivision. Such a convexity, when it exists, is uniquely determined by the corresponding cubical presentation. Some related open problems have been formulated.

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“…G is a gated amalgam of two graphs G 1 and G 2 if G 1 and G 2 are (isomorphic to) two intersecting gated subgraphs of G whose union is all of G. A graph with at least two vertices is said to be prime if it is neither a proper Cartesian product nor a gated amalgam of smaller graphs. For instance, the only prime median graph is the two-vertex complete graph K 2 ; see Isbell [21] and van de Vel [26]. More generally, the prime quasi-median graphs are exactly the complete graphs; quasi-median graphs were introduced by Mulder [23] and further studied in [8,14,28].…”

Section: Academic Pressmentioning

confidence: 99%