Changing and unchanging the diameter of a hypercube

Graham, Niall; Harary, Frank

doi:10.1016/0166-218x(92)90137-y

Cited by 23 publications

(12 citation statements)

References 5 publications

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“…Two vertices are adjacent if they differ only in one bit position. Graham and Harary [4] considered changing the diameter without considering the extent of the change, i.e., they considered D À1 ðGÞ and D þ1 ðGÞ. They showed that D À1 ðQ n Þ ¼ 2, D þ1 ðQ n Þ ¼ n À 1 and D þ0 ðQ n Þ P ðn À 3Þ2 nÀ1 þ 2.…”

Section: Definitions and Propertiesmentioning

confidence: 99%

“…In fact, the diameter of a graph can be affected by adding or deleting edges [1,4,5]. For example, let m-cycle C m be a graph with vertex set f0; 1; 2; .…”

Section: Definitions and Propertiesmentioning

confidence: 99%

“…They concerned whether the diameter is changed or remains unchanged when the edges are increased or decreased. In this paper, we modify three measures proposed in Graham and Harary (1992) to include the extent of the change of the diameter. Let D Àk ðGÞ is the least number of edges whose addition to G decreases the diameter by (at least) k, D þ0 ðGÞ is the maximum number of edges whose deletion from G does not change the diameter, and D þk ðGÞ is the least number of edges whose deletion from G increases the diameter by (at least) k. In this paper, we find the values of D Àk ðC m Þ, D À1 ðT m;n Þ, D À2 ðT m;n Þ, D þ1 ðT m;n Þ, and a lower bound for D þ0 ðT m;n Þ where C m be a cycle with m vertices, T m;n be a torus of size m by n.…”

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confidence: 99%

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Diameter variability of cycles and tori

Wang

Ferrero

et al. 2008

Information Sciences

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Section: Definitions and Propertiesmentioning

confidence: 99%

“…In fact, the diameter of a graph can be affected by adding or deleting edges [1,4,5]. For example, let m-cycle C m be a graph with vertex set f0; 1; 2; .…”

Section: Definitions and Propertiesmentioning

confidence: 99%

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confidence: 99%

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Diameter variability of cycles and tori

Wang

Ferrero

et al. 2008

Information Sciences

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“…Therefore, perhaps some, or even most of the links can be removed from Q n without increasing its diameter. Discovering the number of removable links has been a subject of recent research [16,10,14]. It has been shown that (n − 2)2…”

Section: Progress In Finding Factors Of Small Diametermentioning

confidence: 99%

Hamilton Decompositions and (n/2)-Factorizations of Hypercubes

Bass¹,

Sudborough²

2003

JGAA

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Since Qn, the hypercube of dimension n, is known to have n linkdisjoint paths between any two nodes, the links of Qn can be partitioned into multiple link-disjoint spanning subnetworks, or factors. We seek to identify factors which efficiently simulate Qn, while using only a portion of the links of Qn. We seek to identify (n/2)-factorizations, of Qn where 1) the factors have as small a diameter as possible, and 2) mappings (embeddings) of Qn to each of the factors exist, such that the maximum number of links in a factor corresponding to one link in Qn (dilation), is as small as possible. In this paper we consider two algorithms for generating Hamilton decompositions of Qn, and three methods for constructing (n/2)-factorizations of Qn for specific values of n. The most notable (n/2)-factorization of Qn results in two mutually isomorphic factors, each with diameter n + 2, where an embedding exists which maps Qn to each of the factors with constant dilation.

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Section: Progress In Finding Factors With Small Diametermentioning

confidence: 99%

Symmetric k-factorizations of hypercubes with factors of small diameter

Bass

Sudborough

Proceedings International Symposium on Parallel Architectures, Algorithms and Networks. I-Span'02

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The links of the hypercube Q n can be partitioned into multiple link-disjoint spanning subnetworks, or factors. Each of these factors could simulate Q n . We therefore identify k-factorizations, or partitions of the links of Q n into factors of degree k, where 1) the factorization exists for all values of n such that n mod k = 0, 2) k is as small as possible, 3) the n/k factors have a similar structure, 4) the factors have as small a diameter as possible, and 5) the factors host Q n with as small a dilation as possible.In this paper, we give an (n/2)-factorization of Q n , where n is even, generated by variations on reduced and thin hypercubes. The two factors are isomorphic, and both of the factors have diameter n+2. The diameter is an improvement over the best result known. Both of the factors also host Q n with Θ(1) dilation.

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Changing and unchanging the diameter of a hypercube

Cited by 23 publications

References 5 publications

Diameter variability of cycles and tori

Diameter variability of cycles and tori

Hamilton Decompositions and (n/2)-Factorizations of Hypercubes

Symmetric k-factorizations of hypercubes with factors of small diameter

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