The generalized 4-connectivity of hypercubes

Lin, Shangwei; Zhang, Qianhua

doi:10.1016/j.dam.2016.12.003

Cited by 50 publications

(12 citation statements)

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“…There is an edge between two vertices whenever their binary string representation differs in only one bit position. The dual-cube was introduced by Li and Peng in [14]. As an invariant of the hypercube, it not only keeps numerous desirable properties of the hypercube, but also reduces the interconnection complexity.…”

Section: Preliminarymentioning

confidence: 99%

“…In addition, there are some results of the generalized k-connectivity for some classes of graphs and most of them are about k = 3. For example, Chartrand et al [4] studied the generalized connectivity of complete graphs; Li et al [6] first studied the generalized 3-connectivity of Cartesian product graphs, then Li et al [11] studied the generalized 3connectivity of graph products; Li et al [13] studied the generalized connectivity of the complete bipartite graphs and Lin et al [14] studied the generalized 4-connectivity of hypercubes. As the Cayley graph has some attractive properties to design interconnection networks, Li et al [9] studied the generalized 3-connectivity of star graphs and bubble-sort graphs and Li et al [7] studied the generalized 3-connectivity of the Cayley graph generated by trees and cycles.…”

Section: Introductionmentioning

confidence: 99%

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Two kinds of generalized connectivity of dual cubes

Zhao

Hao

Cheng

2019

Discrete Applied Mathematics

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Let S ⊆ V (G) and κ G (S) denote the maximum number k of edge-disjoint trees T 1 , T 2 , · · · , T k in G such that V (T i ) V (T j ) = S for any i, j ∈ {1, 2, · · · , k} and i = j. For an integer r with 2 ≤ r ≤ n, the generalized r-connectivity of a graph G is defined as κ r (G) = min{κ G (S)|S ⊆ V (G) and |S| = r}. The r-component connectivity cκ r (G) of a non-complete graph G is the minimum number of vertices whose deletion results in a graph with at least r components. These two parameters are both generalizations of traditional connectivity. Except hypercubes and complete bipartite graphs, almost all known κ r (G) are about r = 3. In this paper, we focus on κ 4 (D n ) of dual cube D n . We first show that κ 4 (D n ) = n − 1 for n ≥ 4. As a corollary, we obtain κ 3 (D n ) = n − 1 for n ≥ 4. Furthermore, we show that cκ r+1 (D n ) = rn − r(r+1) 2 + 1 for n ≥ 2 and 1 ≤ r ≤ n − 1.

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Section: Preliminarymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Two kinds of generalized connectivity of dual cubes

Zhao

Hao

Cheng

2019

Discrete Applied Mathematics

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“…Moreover, k n (G) is exactly the spanning tree packing number of G. Therefore, the generalized connectivity is a common generalization of the classical connectivity and spanning tree packing number. The research about S−Steiner trees, spanning tree packing number, generalized connectivity and pendant tree connectivity of graphs plays a key role in effective information transportation in terms of parallel routing design for large-scale networks, see [2,3,4,6,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,27,28,29,30].…”

Section: Introductionmentioning

confidence: 99%

Pendant 3-tree Connectivity of Augmented Cubes

Mane,

Kandekar

2021

Preprint

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The Steiner tree problem in graphs has applications in network design or circuit layout. Given a set S of vertices, |S| ≥ 2, a tree connecting all vertices of S is called an S-Steiner tree (tree connecting S). The reliability of a network G to connect any S vertices (|S| number of vertices) in G can be measure by this parameter. For an S-Steiner tree, if the degree of each vertex in S is equal to one, then that tree is called a pendant S-Steiner tree. Two pendant S-Steiner trees T and T ′ are said to be internally disjoint ifThe local pendant tree-connectivity τG(S) is the maximum number of internally disjoint pendant S-Steiner trees in G. For an integer k with 2 ≤ k ≤ n, the pendant k-tree-connectivity is defined as τ k (G) = min{τG(S) : S ⊆ V (G), |S| = k}. In this paper, we study the pendant 3tree connectivity of Augmented cubes which are modification of hypercubes invented to increase the connectivity and decrease the diameter hence superior to hypercubes. We show that τ3(AQn) = 2n − 3. , which attains the upper bound of τ3(G) given by Hager, for G = AQn.

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“…In addition, there are some results about the generalized k-connectivity of some classes of graphs and most of them are about k = 3. For example, Chartrand et al [3] studied the generalized connectivity of complete graphs; Li et al [14] characterized the minimally 2-connected graphs G with generalized connectivity κ 3 (G) = 2; Li et al [6] studied the generalized 3-connectivity of Cartesian product graphs; Li et al [12] studied the generalized 3-connectivity of graph products; Li et al [15] studied the generalized connectivity of the complete bipartite graphs; Li et al [10] studied the generalized 3-connectivity of the star graphs and bubble-sort graphs; Li et al [8] studied the generalized 3-connectivity of the Cayley graph generated by trees and cycles and Lin and Zhang [16] studied the generalized 4-connectivity of hypercubes. In addition, there are many interesting results about the generalized connectivity can be find in the book [7].…”

Section: Introductionmentioning

confidence: 99%

The Generalized Connectivity of Data Center Networks

Chen

Yang

2019

Parallel Process. Lett.

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The generalized [Formula: see text]-connectivity of a graph [Formula: see text] is a parameter that can measure the reliability of a network [Formula: see text] to connect any [Formula: see text] vertices in [Formula: see text], which is a generalization of traditional connectivity. Let [Formula: see text] and [Formula: see text] denote the maximum number [Formula: see text] of edge-disjoint trees [Formula: see text] in [Formula: see text] such that [Formula: see text] for any [Formula: see text] and [Formula: see text]. For an integer [Formula: see text] with [Formula: see text], the generalized [Formula: see text]-connectivity of a graph [Formula: see text] is defined as [Formula: see text] and [Formula: see text]. Data centers are essential to the business of companies such as Google, Amazon, Facebook and Microsoft et al. Based on data centers, the data center networks [Formula: see text], introduced by Guo et al. in 2008, have many desirable properties. In this paper, we study the generalized [Formula: see text]-connectivity of [Formula: see text] and show that [Formula: see text] for [Formula: see text] and [Formula: see text].

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The generalized 4-connectivity of hypercubes

Cited by 50 publications

References 23 publications

Two kinds of generalized connectivity of dual cubes

Two kinds of generalized connectivity of dual cubes

Pendant 3-tree Connectivity of Augmented Cubes

The Generalized Connectivity of Data Center Networks

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