Conditions for invariance of set diameters under d-convexification in a graph

Soltan, Valeriu; Chepoi, Vo D.

doi:10.1007/bf01068561

Cited by 61 publications

(38 citation statements)

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“…Theorem 4.5 [33,53]. A graph G is bridged if and only if for any convex subset S of G and any integer k ≥ 0 the k-neighborhood B k S is convex.…”

Section: Theorem 43 [9] a Graph G Is Hereditary Modular If And Onlymentioning

confidence: 98%

“…A cycle C of a graph G is well bridged [33] if, for each vertex x of C either the two neighbors u v of x in C are adjacent, or d G x y < d C x y for some antipode y of x in C. In a bridged graph G all cycles C are well bridged [33,53]. Indeed, if the length of C is 2k y is the antipode of x in C, and d G x y = k then from the convexity of the ball B k−1 y we infer that u v are adjacent.…”

Section: Theorem 43 [9] a Graph G Is Hereditary Modular If And Onlymentioning

confidence: 98%

“…An important class of hereditary weakly modular graphs is formed by bridged graphs. A graph is called bridged if all isometric cycles of G have length three; that is, each cycle of length greater than 3 has a shortcut in G see [33,53].…”

Section: Theorem 43 [9] a Graph G Is Hereditary Modular If And Onlymentioning

confidence: 99%

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Graphs of Some CAT(0) Complexes

Chepoi

2000

Advances in Applied Mathematics

215

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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“…Theorem 4.5 [33,53]. A graph G is bridged if and only if for any convex subset S of G and any integer k ≥ 0 the k-neighborhood B k S is convex.…”

Section: Theorem 43 [9] a Graph G Is Hereditary Modular If And Onlymentioning

confidence: 98%

Section: Theorem 43 [9] a Graph G Is Hereditary Modular If And Onlymentioning

confidence: 98%

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Graphs of Some CAT(0) Complexes

Chepoi

2000

Advances in Applied Mathematics

215

251

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“…Fact 2 [11,14]. A graph G is bridged if and only if for every convex set S of G and every integer k the set B k ðSÞ is convex.…”

Section: Auxiliary Factsmentioning

confidence: 99%

On covering bridged plane triangulations with balls

Chepoi

Vaxès

2003

Journal of Graph Theory

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“…[7,10]. A graph is called bridged if G does not contain any isometric cycle of length greater than 3, that is, each cycle of length greater than 3 has a shortcut in G; see Soltan and Chepoi [25] and Farber and Jamison [20]. Bridged graphs can easily be constructed since, according to [1, Corollaries 2.4 and 2.6], they admit certain vertex elimination schemes (relaxing simplicial elimination for chordal graphs).…”

Section: Academic Pressmentioning

confidence: 99%