Spherical and Clockwise Spherical Graphs

Berrachedi, Abdelhafid; Havel, Ivan M.; Mulder, Henry Martyn

doi:10.1023/a:1026227101941

Cited by 4 publications

(3 citation statements)

References 7 publications

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“…Axiom (A5) is a weak variant of the sphericity condition studied in [52]. In the case of rank n = 2, this axiom together with (A1) and specifying n = 2 in (A2) then describes the generalized quadrangles [146].…”

Section: Meshed Graphs and Weak Modularitymentioning

confidence: 99%

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Metric graph theory and geometry: a survey

Bandelt¹,

Chepoi²

2008

Contemporary Mathematics

144

155

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The article surveys structural characterizations of several graph classes defined by distance properties, which have in part a general algebraic flavor and can be interpreted as subdirect decomposition. The graphs we feature in the first place are the median graphs and their various kinds of generalizations, e.g., weakly modular graphs, or fiber-complemented graphs, or l 1-graphs. Several kinds of l 1-graphs admit natural geometric realizations as polyhedral complexes. Particular instances of these graphs also occur in other geometric contexts, for example, as dual polar graphs, basis graphs of (even ∆-)matroids, tope graphs, lopsided sets, or plane graphs with vertex degrees and face sizes bounded from below. Several other classes of graphs, e.g., Helly graphs (as injective objects), or bridged graphs (generalizing chordal graphs), or tree-like graphs such as distance-hereditary graphs occur in the investigation of graphs satisfying some basic properties of the distance function, such as the Helly property for balls, or the convexity of balls or of the neighborhoods of convex sets, etc. Operators between graphs or complexes relate some of the graph classes reported in this survey.

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Section: Meshed Graphs and Weak Modularitymentioning

confidence: 99%

“…Every interval I (u, v) in such a graph G necessarily is the covering graph of a complemented modular lattice (with bounds u and v). Therefore G is weakly spherical [52] in the sense that for every vertex x between two vertices u and v there exists some vertex x such that v, x, u, x form a metric rectangle.…”

Section: Modular Graphs and Orientabilitymentioning

confidence: 99%

Metric graph theory and geometry: a survey

Bandelt¹,

Chepoi²

2008

Contemporary Mathematics

144

155

View full text Add to dashboard Cite

show abstract

“…Because G is connected, the result follows. As noted in [2] and [7], a graph G is a hypercube if and only if G is spherical and bipartite. We aim to substitute this condition for a graph to be spherical with a weaker condition of antipodality and with a local condition for a graph to be a (0, 2)-graph.…”

Section: Bipartite and Antipodal Graphsmentioning

confidence: 99%