Another characterization of hypercubes

Laborde, Jean-Marie; Hebbare, S. P. Rao

doi:10.1016/0012-365x(82)90139-x

Cited by 24 publications

(12 citation statements)

References 2 publications

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“…A similar but dierent condition (see criterion C2) was derived by Garey and Graham [6] in their study of``squashed cubes''. Combining their results with the Merger's Theorem [7, p. 47, Theorem 5.9], we easily conclude that the number of node-disjoint u±v paths in a Q n is d. The third condition is due to Laborde and Hebbare [12].…”

Section: Introductionmentioning

confidence: 79%

“…We next list characterizations for a given graph G to be a hypercube Q n [5,6,8,12]. In principle, each of these conditions contains enough information to enable the logical deduction that G is indeed an n-cube graph or n-dimensional hypercube (we shall only list these criteria and include a reference for each, where the details may be found).…”

Section: Introductionmentioning

confidence: 99%

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Optimal Gray-code labeling and recognition algorithms for hypercubes

Nikolopoulos

2001

Information Sciences

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We present an optimal greedy algorithm which returns a Gray-code labeling of the nodes of an n-dimensional hypercube; that is, a labeling of the nodes with binary strings of length n for which the Hamming distance between two nodes is 1 if and only if these are adjacent in the hypercube. The proposed algorithm is very simple; it uses breadth®rst search to guide the greedy choice of nodes and computes the Gray-code label of a node u by performing the logical disjunction of the Gray-code labels of two nodes adjacent to node u. It takes as input a hypercube Q n with N 2 n nodes and runs in ON log N time. Based on the labeling algorithm we propose a recognition algorithm for hypercubes which runs in ON log N time. Thus, in view of the fact that Q n has n2 nÀ1 edges, this behaviour is optimal. Both labeling and recognition algorithms incorporate such algorithmic features that they can be optimally implemented in a PRAM model of computation. Ó

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Section: Introductionmentioning

confidence: 79%

Section: Introductionmentioning

confidence: 99%

Optimal Gray-code labeling and recognition algorithms for hypercubes

Nikolopoulos

2001

Information Sciences

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“…We use the fact that in the hypercube, any two vertices at distance 2 have exactly 2 neighbors in common (cf. [7,9]), and prove by induction the following statement:…”

Section: Resultsmentioning

confidence: 99%

Spanning caterpillars of a hypercube

Dvo?�k¹,

Havel²,

Laborde³

et al. 1997

J. Graph Theory

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“…by making use of its adjacency matrix. The new characterization is based on the results in [29] where the authors characterized the hypercube in a certain space based on the relations between distinct adjacent edges of a graph. For notational convenience, we introduce the following definition.…”

Section: Theorem 34 Let W E^i"'"'''^ Be the Cubic Matrix Defined By mentioning

confidence: 99%

“…In particular, Laborde et'al [29] observed the following interesting link between an exact 4-cycle graph and the hypercube in a certain space. Theorem 3.6 [29] Suppose that a graph G = (V,£) is a connected exact 4-cycle graph. If the degree of every vertex ofG is d and \V\ = 2 , then G is a hypercube.…”

Section: Definition 35 a Graph G = (V£) Is Called An Exact 4-cycle mentioning

confidence: 99%

Graph Modeling for Quadratic Assignment Problems Associated with the Hypercube

Mittelmann¹,

Peng²,

Wu³

2009

AIP Conference Proceedings

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In the paper we consider the quadratic assignment problem arising from channel coding in communications where one coefficient matrix is the adjacency matrix of a hypercube in a finite dimensional space. By using the geometric structure of the hypercube, we first show that there exist at least n different optimal solutions to the underlying QAPs. Moreover, the inherent symmetries in the associated hypercube allow us to obtain partial information regarding the optimal solutions and thus shrink the search space and improve all the existing QAP solvers for the underlying QAPs.Secondly, we use graph modeling technique to derive a new integer linear program (ILP) models for the underlying QAPs. The new ILP model has n{n -1) binary variables and 0{n log(n)) linear constraints. This yields the smallest known number of binary variables for the ILP reformulation of QAPs. Various relaxations of the new ILP model are obtained based on the graphical characterization of the hypercube, and the lower bounds provided by the LP relaxations of the new model are analyzed and compared with what provided by several classical LP relaxations of QAPs in the literature.

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Another characterization of hypercubes

Cited by 24 publications

References 2 publications

Optimal Gray-code labeling and recognition algorithms for hypercubes

Optimal Gray-code labeling and recognition algorithms for hypercubes

Spanning caterpillars of a hypercube

Graph Modeling for Quadratic Assignment Problems Associated with the Hypercube

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