Two characterizations of generalized hypercube

Mollard, Michel

doi:10.1016/0012-365x(91)90217-p

Cited by 6 publications

(4 citation statements)

References 7 publications

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“…Later, [Mollard(1991)] generalized Proposition 1 to arbitrary Hamming graphs. For any vertex x in the graph G let N i (x) denote the number of maximal i-cliques K i in G that contain the vertex x.…”

Section: Hamming Graphs and Their Geodesic Intervalsmentioning

confidence: 99%

“…Proposition 2 ( [ Mollard(1991)]) Let G be a simple connected graph such that two non-adjacent vertices in G either have exactly 2 common neighbors or none at all, and suppose G has neither K 4 \ e nor K 2 K 3 \ e ( Figure 2) as induced subgraph. Then N i (x) is independent of x and G is isomorphic to the Hamming graph if and only if |V (G)| = p h=1 h Ni(x) , where p is the maximum integer such that N p (x) is nonzero.…”

Section: Hamming Graphs and Their Geodesic Intervalsmentioning

confidence: 99%

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Transit sets of -point crossover operators

Changat

Narasimha-Shenoi²,

Nezhad

et al. 2020

AKCE International Journal of Graphs and Combinatorics

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k-point crossover operators and their recombination sets are studied from different perspectives. We show that transit functions of k-point crossover generate, for all k > 1, the same convexity as the interval function of the underlying graph. This settles in the negative an open problem by Mulder about whether the geodesic convexity of a connected graph G is uniquely determined by its interval function I. The conjecture of Gitchoff and Wagner that for each transit set R k (x, y) distinct from a hypercube there is a unique pair of parents from which it is generated is settled affirmatively. Along the way we characterize transit functions whose underlying graphs are Hamming graphs, and those with underlying partial cube graphs. For general values of k it is shown that the transit sets of k-point crossover operators are the subsets with maximal Vapnik-Chervonenkis dimension. Moreover, the transit sets of k-point crossover on binary strings form topes of uniform oriented matroid of VC-dimension k + 1. The Topological Representation Theorem for oriented matroids therefore implies that k-point crossover operators can be represented by pseudosphere arrangements. This provides the tools necessary to study the special case k = 2 in detail.

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Section: Hamming Graphs and Their Geodesic Intervalsmentioning

confidence: 99%

Section: Hamming Graphs and Their Geodesic Intervalsmentioning

confidence: 99%

Transit sets of -point crossover operators

Changat

Narasimha-Shenoi²,

Nezhad

et al. 2020

AKCE International Journal of Graphs and Combinatorics

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“…For different characterizations of these graphs see [2,3,21,22,23]. The special case when all factor graphs are of the same order is treated in [9].…”

Section: Introductionmentioning

confidence: 99%

Characterizing subgraphs of Hamming graphs

Klavžar

Peterin

2005

Journal of Graph Theory

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Cartesian products of complete graphs are known as Hamming graphs. Using embeddings into Cartesian products of quotient graphs we characterize subgraphs, induced subgraphs, and isometric subgraphs of Hamming graphs. For instance, a graph G is an induced subgraph of a Hamming graph if and only if there exists a labeling of E(G) fulfilling the following two conditions: (i) edges of a triangle receive the same label; (ii) for any vertices u and v at distance at least two, there exist two labels which both appear on any induced u; v-path.

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“…They play an important role in Communication and Coding theory and other fields by their extensive applications [7] as well as in Graph Theory by their very nice combinatorial and metric properties [3,5]. They have been characterized in several ways (see [2,4,8]), involving especially the notions of interval and distance functions or some related concepts. Among these, one can mention the characterization due to Mollard [8], where the notions of quasidistance monotonicity and quasi-interval monotonicity are introduced and used.…”

Section: Introductionmentioning

confidence: 99%

Distance monotonicity and a new characterization of Hamming graphs

Aouchiche

2005

Information Processing Letters

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Two characterizations of generalized hypercube

Cited by 6 publications

References 7 publications

Transit sets of -point crossover operators

Transit sets of -point crossover operators

Characterizing subgraphs of Hamming graphs

Distance monotonicity and a new characterization of Hamming graphs

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