On Isometric Embeddings of Graphs

Graham, R. L.; Winkler, P. M.

doi:10.2307/1999951

Cited by 30 publications

(42 citation statements)

References 4 publications

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“…Another more general algorithm for recognizing partial Hamming graphs (isometric subgraphs of Cartesian products of complete graphs) of complexity O(mn) is given in [1]. Applying the canonical isometric embedding theory of Graham and Winkler [14], a simple algorithm for recognizing partial Hamming graphs of the same complexity can be obtained; see [17,19]. However, only a trivial lower bound O(m) for recognizing partial cubes is known.…”

Section: Introductionmentioning

confidence: 99%

Partial Cubes and Crossing Graphs

Klavžar¹,

Mulder²

2002

SIAM J. Discrete Math.

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Abstract. Partial cubes are defined as isometric subgraphs of hypercubes. For a partial cube G, its crossing graph G # is introduced as the graph whose vertices are the equivalence classes of the Djoković-Winkler relation Θ, two vertices being adjacent if they cross on a common cycle. It is shown that every graph is the crossing graph of some median graph and that a partial cube G is 2-connected if and only if G # is connected. A partial cube G has a triangle-free crossing graph if and only if G is a cube-free median graph. This result is used to characterize the partial cubes having a tree or a forest as its crossing graph. An expansion theorem is given for the partial cubes with complete crossing graphs. Cartesian products are also considered. In particular, it is proved that G # is a complete bipartite graph if and only if G is the Cartesian product of two trees.

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Section: Introductionmentioning

confidence: 99%

Partial Cubes and Crossing Graphs

Klavžar¹,

Mulder²

2002

SIAM J. Discrete Math.

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“…The following theorem combines some important results mentioned in [Men54,HW88,GW85,Gow85]. Theorem 1.2.…”

Section: Introductionmentioning

confidence: 53%

Distance geometry for kissing spheres

Chen

2015

Linear Algebra and its Applications

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“…Since Γ projects surjectively onto both Γ 1 and Γ 2 , we can now determine the Graham-Winkler Cartesian product graph for Γ (cf. [12]). Namely, that Cartesian product graph has d complete graphs of size two and the cocktail-party graph Γ 2 as its factors.…”

Section: The Skeleton Ofmentioning

confidence: 99%

Hypercube embedding of Wythoffians

2008

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The Wythoff construction takes a d-dimensional polytope P , a subset S of {0, . . . , d} and returns another d-dimensional polytope P (S). If P is a regular polytope, then P (S) is vertex-transitive. This construction builds a large part of the Archimedean polytopes and tilings in dimension 3 and 4.We want to determine, which of those Wythoffians P (S) with regular P have their skeleton or dual skeleton isometrically embeddable into the hypercubes H m and half-cubes 1 2 H m . We find six infinite series, which, we conjecture, cover all cases for dimension d > 5 and some sporadic cases in dimension 3 and 4 (see Tables 1 and 2).Three out of those six infinite series are explained by a general result about the embedding of Wythoff construction for Coxeter groups. In the last section, we consider the Euclidean case; also, zonotopality of embeddable P (S) are addressed throughout the text.

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On Isometric Embeddings of Graphs

Cited by 30 publications

References 4 publications

Partial Cubes and Crossing Graphs

Partial Cubes and Crossing Graphs

Distance geometry for kissing spheres

Hypercube embedding of Wythoffians

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