The generalized 3-connectivity of graph products

Li, Hengzhe; Ma, Yingbin; Yang, Weihua; Wang, Yifei

doi:10.1016/j.amc.2016.10.002

Cited by 32 publications

(14 citation statements)

References 6 publications

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“…By Lemma 2, κ(D 01 ) = κ(D 11 ) = n−1. Then there are n−1 internally disjoint paths P 1 , P 2 , · · · , P n−1 between x and y in D 01 and n−1 internally disjoint paths P ′ 1 , P ′ 2 , · · · , P ′ n−1 between z and w in D 11 . Let x i ∈ V (P i ) such that x 1 i , the outside neighbour of x i , does not belong to D 11 .…”

Section: Proof Without Loss Of Generality Let Hmentioning

confidence: 99%

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: Proof Without Loss Of Generality Let Hmentioning

confidence: 99%

Two kinds of generalized connectivity of dual cubes

Zhao

Hao

Cheng

2019

Discrete Applied Mathematics

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“…Also in [11], Li et al proposed a conjecture as follows: 11]). Let G be a nontrivial connected graph, and let H be a 3connected graph.…”

Section: Theorem 12 ([11]mentioning

confidence: 99%

“…Later in [11], Li et al gave a better result when H becomes a 2-connected graph. Theorem 1.2 ([11]).…”

mentioning

confidence: 99%

Two lower bounds for generalized 3-connectivity of Cartesian product graphs

Gao

Wang

2018

Applied Mathematics and Computation

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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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The generalized 3-connectivity of graph products

Cited by 32 publications

References 6 publications

Two kinds of generalized connectivity of dual cubes

Two kinds of generalized connectivity of dual cubes

Two lower bounds for generalized 3-connectivity of Cartesian product graphs

Pendant 3-tree Connectivity of Augmented Cubes

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