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Metric graph theory and geometry: a survey
 CONTEMPORARY MATHEMATICS
"... 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 general ..."
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Cited by 32 (8 self)
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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 fibercomplemented graphs, or l1graphs. Several kinds of l1graphs 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 treelike graphs such as distancehereditary 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
Applications of Cut Polyhedra
, 1992
"... We group in this paper, within a unified framework, many applications of the following polyhedra: cut, boolean quadric, hypermetric and metric polyhedra. We treat, in particular, the following applications: ffl ` 1  and L 1 metrics in functional analysis, ffl the maxcut problem, the Boole probl ..."
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Cited by 23 (2 self)
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We group in this paper, within a unified framework, many applications of the following polyhedra: cut, boolean quadric, hypermetric and metric polyhedra. We treat, in particular, the following applications: ffl ` 1  and L 1 metrics in functional analysis, ffl the maxcut problem, the Boole problem and multicommodity flow problems in combinatorial optimization, ffl lattice holes in geometry of numbers, ffl density matrices of manyfermions systems in quantum mechanics. We present some other applications, in probability theory, statistical data analysis and design theory.
Hypermetrics in Geometry of Numbers
, 1993
"... . A finite semimetric d on a set X is hypermetric if it satisfies the inequality P i;j2X b i b j d ij 0 for all b 2 Z X with P i2X b i = 1. Hypermetricity turns out to be the appropriate notion for describing the metric structure of holes in lattices. We survey hypermetrics, their connection ..."
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Cited by 11 (3 self)
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. A finite semimetric d on a set X is hypermetric if it satisfies the inequality P i;j2X b i b j d ij 0 for all b 2 Z X with P i2X b i = 1. Hypermetricity turns out to be the appropriate notion for describing the metric structure of holes in lattices. We survey hypermetrics, their connections with lattices and applications. 2 M. Deza, V.P. Grishukhin and M. Laurent Contents 1 Introduction 2 Preliminaries 2.1 Distance spaces Metric notions Operations on distance spaces Preliminary results on distance spaces 2.2 Lattices and Lpolytopes Lattices Lpolytopes Lpolytopes and Voronoi polytopes Lattices and positive quadratic forms Lpolytopes and empty ellipsoids Basic facts on Lpolytopes Construction of Lpolytopes Lpolytopes in dimension k 4 2.3 Finiteness of the number of types of Lpolytopes in given dimension 3 Hypermetrics and Lpolytopes 3.1 The connection between hypermetrics and Lpolytopes 3.2 Polyhedrality of the hypermetric cone 3.3 Lpolytopes in root lattic...
Partial cubes: structures, characterizations, and constructions
 Discrete Math
"... Partial cubes are isometric subgraphs of hypercubes. Structures on a graph defined by means of semicubes, and Djoković’s and Winkler’s relations play an important role in the theory of partial cubes. These structures are employed in the paper to characterize bipartite graphs and partial cubes of arb ..."
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Cited by 7 (2 self)
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Partial cubes are isometric subgraphs of hypercubes. Structures on a graph defined by means of semicubes, and Djoković’s and Winkler’s relations play an important role in the theory of partial cubes. These structures are employed in the paper to characterize bipartite graphs and partial cubes of arbitrary dimension. New characterizations are established and new proofs of some known results are given. The operations of Cartesian product and pasting, and expansion and contraction processes are utilized in the paper to construct new partial cubes from old ones. In particular, the isometric and lattice dimensions of finite partial cubes obtained by means of these operations are calculated. Key words: Hypercube, partial cube, semicube 1
Tribes of Cubic Partial Cubes
 Discrete Appl. Math
, 2005
"... Partial cubes are graphs isometrically embeddable into hypercubes. Three infinite families and a few sporadic examples of cubic partial cubes are known. ..."
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Cited by 4 (1 self)
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Partial cubes are graphs isometrically embeddable into hypercubes. Three infinite families and a few sporadic examples of cubic partial cubes are known.
The Classification of Finite Connected Hypermetric Spaces
 GRAPHS AND COMBINATORICS
, 1987
"... A finite distance space X,d d:X 2,7 / is hypermetric (of negative type) if axay d(x, y) < 0 for all integral sequences {axlx ~ X} that sum to 1 (sum to 0). X, d is connected if the set {(x, y)ld(x, y) = 1, x, y ~ X} is the edge set for a connected graph on X, and graphical if d is the path l ..."
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Cited by 3 (2 self)
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A finite distance space X,d d:X 2,7 / is hypermetric (of negative type) if axay d(x, y) < 0 for all integral sequences {axlx ~ X} that sum to 1 (sum to 0). X, d is connected if the set {(x, y)ld(x, y) = 1, x, y ~ X} is the edge set for a connected graph on X, and graphical if d is the path length distance for this graph. Then we prove Theorem 1. A connected space X, d has negative type if and only if X may be realised as a subset of a Euclidean space E, IJ II, sdch that (i) X contains 0 and spans E (ii} d(x,y) = 1/2tlx Yll 2 (x,y~S) (iii) L = Y_X is a root lattice, i.e. an orthogonal direct sum of lattices of type A,, D,, E6, ET, and Es. Call a hypermetric space X, d complete if for each triple x, y, z ~ X with d(y, z) = 1 and d(x, y) + 1 =
Fullerenes and Coordination Polyhedra versus HalfCubes Embeddings
, 1997
"... A fullerene F n is a 3regular (or cubic) polyhedral carbon molecule for which the n vertices  the carbons atoms  are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking fo ..."
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A fullerene F n is a 3regular (or cubic) polyhedral carbon molecule for which the n vertices  the carbons atoms  are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking for the list of such fullerenes, we first check the embeddability of all fullerenes F n for n ! 60 and of all preferable fullerenes C n for n ! 86 and their duals. Then, we consider some infinite families, including fullerenes with icosahedral symmetry, which describe virus capsids, onionlike metallic clusters and geodesic domes. Quasiembeddings and fullerene analogues are considered. We also present some results on chemically relevant polyhedra such as coordination polyhedra and cluster polyhedra. Finally we conjecture that the list of known embeddable fullerenes is complete and present its relevance to the Katsura model for vesicles cells. Contents 1 Introduction and Basic Properties 2 1...
EdgeCritical Isometric Subgraphs of Hypercubes
, 2001
"... Isometric subgraphs of hypercubes are known as partial cubes. Edgecritical partial cubes are introduced as the partial cubes G for which Ge is not a partial cube for any edge e of G. An expansion theorem is proved by means of which one can generate many edgecritical partial cubes. Edgecritical p ..."
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Isometric subgraphs of hypercubes are known as partial cubes. Edgecritical partial cubes are introduced as the partial cubes G for which Ge is not a partial cube for any edge e of G. An expansion theorem is proved by means of which one can generate many edgecritical partial cubes. Edgecritical partial cubes are characterized among the Cartesian product graphs. We also show that the 3cube and the subdivision graph of K[4] are the only edgecritical partial cubes on at most 10 vertices.
Hypercube Embeddings and Designs
, 1994
"... This is a survey on hypercube embeddable semimetrics and the link with designs. We investigate, in particular, the variety of hypercube embeddings of the equidistant metric. For some parameters, it is linked with the question of existence of projective planes or Hadamard matrices. The problem of tes ..."
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This is a survey on hypercube embeddable semimetrics and the link with designs. We investigate, in particular, the variety of hypercube embeddings of the equidistant metric. For some parameters, it is linked with the question of existence of projective planes or Hadamard matrices. The problem of testing whether a semimetric is hypercube embeddable is NPhard in general. Several classes of semimetrics are described for which this problem can be solved in polynomial time. We also consider questions related to some necessary conditions for hypercube embeddability.
Hypermetric twodistance spaces
"... Any twodistance space is uniquely up to a multiple represented by a distance dG,t for a graph G such that dG,t(ij) is equal to 1 or t depending on (ij) is an edge or nonedge of G. For a cone CA n of npoint distance spaces, we set tA (G):= max{t: dG,t ∈ CA n}. We consider the cut cone Cutn = CC n, ..."
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Any twodistance space is uniquely up to a multiple represented by a distance dG,t for a graph G such that dG,t(ij) is equal to 1 or t depending on (ij) is an edge or nonedge of G. For a cone CA n of npoint distance spaces, we set tA (G):= max{t: dG,t ∈ CA n}. We consider the cut cone Cutn = CC n, the hypermetric cone Hypn = CH n, and the cone of negative type Negn = CN n. The values of tN (G) (in other terms) are considered by many authors, and are determined by roots of some polynomials. We give bounds on tH (G), and consider some classes of graphs G with. The graphs G a given value of tH (G), especially for tH (G) = 2 and tH (G) = 3 2 with tH (G) = 2 are exactly graphs having the hypermetric truncated distance d ∗ G. 1