Conditional (edge-)fault-tolerant strong Menger (edge) connectivity of folded hypercubes

Cheng, Qi; Li, Pingshan; Xu, Min

doi:10.1016/j.tcs.2018.03.011

Cited by 30 publications

(4 citation statements)

References 10 publications

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“…[41]. Folded hypercubes are of wide current interest in reliability and fault tolerance of interconnection networks, see [9,21,25,42]. In this section we add to the known results on folded hypercubes their wirelength.…”

Section: Embeddings Of Folded Hypercubesmentioning

confidence: 99%

Lower bounds for dilation, wirelength, and edge congestion of embedding graphs into hypercubes

et al. 2020

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Interconnection networks provide an effective mechanism for exchanging data between processors in a parallel computing system. One of the most efficient interconnection networks is the hypercube due to its structural regularity, potential for parallel computation of various algorithms, and the high degree of fault tolerance. Thus it becomes the first choice of topological structure of parallel processing and computing systems. In this paper, lower bounds for the dilation, wirelength, and edge congestion of an embedding of a graph into a hypercube are proved. Two of these bounds are expressed in terms of the bisection width. Applying these results, the dilation and wirelength of embedding of certain complete multipartite graphs, folded hypercubes, wheels, and specific Cartesian products are computed.

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Section: Embeddings Of Folded Hypercubesmentioning

confidence: 99%

Lower bounds for dilation, wirelength, and edge congestion of embedding graphs into hypercubes

et al. 2020

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“…(2n − 2)-conditional edge-fault-tolerant strongly Menger edge connected for n ≥ 5. Cheng et al [6] proved that the n-dimensional folded hypercube is (3n − 5)-conditional edge-faulttolerant strongly Menger edge connected for n ≥ 5. Li et al [12] proved that the n-dimensional balanced hypercube is (2n − 2)-edge-fault-tolerant strongly Menger edge connected and (6n − 8)-conditional edge-fault-tolerant strongly Menger edge connected for n ≥ 2.…”

Section: Introductionmentioning

confidence: 99%

Menger-type connectivity of line graphs of faulty hypercubes

Jia¹,

Qian²

2022

Preprint

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A connected graph G is called strongly Menger edge connected if G has min{deg G (x), deg G (y)} edge-disjoint paths between any two distinct vertices x and y in G. In this paper, we consider two types of strongly Menger edge connectivity of the line graphs of n-dimensional hypercube-like networks with faulty edges, namely the m-edgefault-tolerant and m-conditional edge-fault-tolerant strongly Menger edge connectivity. We show that the line graph of any n-dimensional hypercube-like network is (2n − 4)edge-fault-tolerant strongly Menger edge connected for n ≥ 3 and (4n − 10)-conditional edge-fault-tolerant strongly Menger edge connected for n ≥ 4. The two bounds for the maximum number of faulty edges are best possible.

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“…On one hand, the (conditional) strong Menger connectivity of many known networks were explored in literature, for example, star graph [9,12], hypercubes [13], folded hypercubes [21], hypercube-like networks [16], Cayley graphs generated by transposition trees [17], augmented cubes [2], bubble-sort star graph [1], balanced hypercubes [7] etc. On the other hand, the (conditional) strong Menger edge connectivity were investigated on hypercubes [14], folded hypercubes [3], hypercube-like networks [8], balanced hypercubes [7] etc. He et al [6] and Sabir and Meng [15] presented sufficient conditions for a regular graph to be strongly Menger vertex/edge connected with some restricted conditions.…”

Section: Introductionmentioning

confidence: 99%

“…k − S i . By Lemma 1(3), there are 2k edges between adjacent subgraphsAQ j 2,k and AQ j+1 2,k for j ∈ [k] 0 . Since s c ≤ |S| − 2|I| ≤ 11 − 2 × 4 = 3 < 2k,all H i 's for i ∈ I and all subgraphs (AQ j 2,k − S j )'s for j ∈ J belong to the same component (i.e., H) in AQ 2,k − S. Since s c ≤ 3 and every vertex in AQ i 2,k has four distinct extra neighbors, if a vertex of AQ i 2,k − S i is a singleton, then it must be connected to H. Thus, |V (H)| ≥ |V (AQ 2,k )|.…”

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confidence: 99%

Strong Menger connectedness of augmented $k$-ary $n$-cubes

Gu¹,

Chang²,

Hao³

2019

Preprint

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A connected graph G is called strongly Menger (edge) connected if for any two distinct vertices x, y of G, there are min{deg G (x), deg G (y)} vertex(edge)-disjoint paths between x and y. In this paper, we consider strong Menger (edge) connectedness of the augmented k-ary n-cube AQ n,k , which is a variant of k-ary n-cube Q k n . By exploring the topological proprieties of AQ n,k , we show that AQn,3 for n ≥ 4 (resp. AQ n,k for n ≥ 2 and k ≥ 4) is still strongly Menger connected even when there are 4n − 9 (resp. 4n − 8) faulty vertices and AQ n,k is still strongly Menger edge connected even when there are 4n − 4 faulty edges for n ≥ 2 and k ≥ 3. Moreover, under the restricted condition that each vertex has at least two fault-free edges, we show that AQ n,k is still strongly Menger edge connected even when there are 8n − 10 faulty edges for n ≥ 2 and k ≥ 3. These results are all optimal in the sense of the maximum number of tolerated vertex (resp. edge) faults.

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Conditional (edge-)fault-tolerant strong Menger (edge) connectivity of folded hypercubes

Cited by 30 publications

References 10 publications

Lower bounds for dilation, wirelength, and edge congestion of embedding graphs into hypercubes

Lower bounds for dilation, wirelength, and edge congestion of embedding graphs into hypercubes

Menger-type connectivity of line graphs of faulty hypercubes

Strong Menger connectedness of augmented $k$-ary $n$-cubes

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