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Clin d'Oeil on L_1Embeddable Planar Graphs
, 1996
"... In this note we present some properties of L1embeddable planar garphs. We show that every such graph G has a scale 2 embedding into a hypercube. Further, under some additional conditions we show that for a simple circuit C of G the subgraph H of G bounded by C is also L1embeddable. In many importa ..."
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Cited by 17 (2 self)
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In this note we present some properties of L1embeddable planar garphs. We show that every such graph G has a scale 2 embedding into a hypercube. Further, under some additional conditions we show that for a simple circuit C of G the subgraph H of G bounded by C is also L1embeddable. In many important cases, the length of C is the dimension of the smallest cube in which H has a scale embedding. Using these facts we establish the L1embeddability of a list of planar graphs.
Decomposition and l_1Embedding of Weakly Median Graphs
, 1998
"... . Weakly median graphs, being defined by interval conditions and forbidden induced subgraphs, generalize quasimedian graphs as well as pseudomedian graphs. It is shown that finite weakly median graphs can be decomposed with respect to gated amalgamation and Cartesian multiplication into 5wheel ..."
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Cited by 9 (6 self)
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. Weakly median graphs, being defined by interval conditions and forbidden induced subgraphs, generalize quasimedian graphs as well as pseudomedian graphs. It is shown that finite weakly median graphs can be decomposed with respect to gated amalgamation and Cartesian multiplication into 5wheels, induced subgraphs of hyperoctahedra (alias cocktail party graphs), and 2connected bridged graphs not containing K4 or K1;1;3 as an induced subgraph. As a consequence one obtains that every finite weakly median graph is l 1 embeddable, that is, it embeds as a metric subspace into some R n equipped with the 1norm. In this paper we continue to elaborate on a structure theory of graphs based on two fundamental operations, viz., Cartesian multiplication and gated amalgamation. While Cartesian multiplication is a standard operation, gated amalgamation seems to appear only in the context of median graphs and their generalizations; cf. [4, 6, 8, 23, 27]. An induced subgraph H of a grap...
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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Cited by 2 (0 self)
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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...
Embeddings of Graphs
, 1994
"... In this paper, we survey the metric properties of isometric subgraphs of hypercubes and, more generally, of ℓ1graphs. An ℓ1graph is a graph which is hypercube embeddable, up to scale. In particular, we present several characterizations for hypercube embeddable graphs and a combinatorial algorithm ..."
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In this paper, we survey the metric properties of isometric subgraphs of hypercubes and, more generally, of ℓ1graphs. An ℓ1graph is a graph which is hypercube embeddable, up to scale. In particular, we present several characterizations for hypercube embeddable graphs and a combinatorial algorithm (from [Shp93]) permitting to recognize ℓ1graphs in polynomial time. The link with the metric representation of graphs as Cartesian products (from [GW85]) is also described. In particular, we see how a well known equivalence relation of Djokovic [Djo73], leading to the notion of isometric dimension of a graph, plays a central and unifying role between the various embeddability concepts.