The classification of finite connected hypermetric spaces

Terwilliger, Paul; Deza, Michel

doi:10.1007/bf01788552

Cited by 19 publications

(11 citation statements)

References 11 publications

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“…This result paves the way to a description of l 1 -graphs and, more generally, hypermetric graphs via the canonical embedding. The extra indecomposable factors that come into play when passing from l 1 -graphs to the more general hypermetric graphs involve the Gosset graph G 56 , which is the skeleton of the 7-dimensional Gosset polytope: its 56 vertices are obtained by putting signs in the eight possible ways to the characteristic vectors of the Fano plane; see [94,166] The proof of (b) given in [159] is graph-theoretical and can be adapted to test in time O(#V #E) whether a given graph G = (V, E) is an l 1 -graph and whether it can be isometrically embedded into a half-cube; see [95]. This contrast with the NP-completeness of recognizing l 1 -embeddable finite metrics.…”

Section: Scale Embeddingsmentioning

confidence: 99%

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: Scale Embeddingsmentioning

confidence: 99%

Metric graph theory and geometry: a survey

Bandelt¹,

Chepoi²

2008

Contemporary Mathematics

144

155

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“…The graphs with least eigenvalue > -2 were characterized by Cameron, Goethals, Seidel and Shult [9] in terms of root systems. In the same terms, Terwillinger and Deza [21] Proof. Consider the 2A-gonal inequality (1) in the metric transform (GT)C. The left-hand side of ( 1 ) does not decrease when the nonzero terms are replaced by 2C 's, and the right-hand side of ( 1 ) does not increase when the nonzero terms are replaced by l's.…”

Section: Euclidean Embeddingsmentioning

confidence: 83%

Metric Transforms and Euclidean Embeddings

Deza¹,

Maehara²

1990

Transactions of the American Mathematical Society

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“…The following results on connected hypermetrics and connected negative type distance functions were proved in [19]; they are a specification of the results given in section 3. Furthermore, all L-polytopes in irreducible root lattices and whose center is a point of the dual lattice are described in [19]; in graph terms, they are the Johnson graph for the root lattice An, the halfcube and cocktail party graph (corresponding to the cross polytope fin) for the root lattice Dn, the Gosset graph on 56 vertices in ET, the Schl~fli graph on 27 vertices in E 6. One can check that, among tl~e above graphs, the corresponding hypermetrics do not belong to the cut cone precisely for the last two graphs.…”

Section: Discussionmentioning

confidence: 95%