Fault diameter of product graphs

Xu, Jie; Yang, Chao

doi:10.1016/j.ipl.2007.01.001

Cited by 7 publications

(2 citation statements)

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“…(Our notation here slightly differs from notation used in [20] and [27].) The result was later generalized to graph bundles in [2] and generalized graph products (as defined by [9]) in [28]. Here we show that in most cases of Cartesian graph bundles the bound can indeed be improved to the one claimed in [20].…”

Section: Introductionmentioning

confidence: 66%

Improved upper bounds for vertex and edge fault diameters of Cartesian graph bundles

Erveš

Žerovnik

2015

Discrete Applied Mathematics

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Mixed fault diameter of a graph G, D (a,b) (G), is the maximal diameter of G after deletion of any a vertices and any b edges. Special cases are the (vertex) fault diameter D V a = D (a,0) and the edge fault diameter D E a = D (0,a) . Let G be a Cartesian graph bundle with fibre F over the base graph B. We show that(when the graphs F and B are k F -connected and k B -connected, 0 < a < k F , 0 < b < k B , and provided thatwhen the graphs F and B are k F -edge connected and k B -edge connected, 0 ≤ a < k F , 0 ≤ b < k B , and provided that D E a (F ) ≥ 2 and D E b (B) ≥ 2.

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Section: Introductionmentioning

confidence: 66%

Improved upper bounds for vertex and edge fault diameters of Cartesian graph bundles

Erveš

Žerovnik

2015

Discrete Applied Mathematics

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“…Banic, Erves, and Zerovnik showed that in a

(k + 1)

‐connected graph the k ‐fault‐diameter can exceed the k ‐edge‐fault‐diameter by at most one. For results on the (edge‐)fault‐diameter of Cartesian products of graphs and their generalisations see, for example, .…”

Section: Introductionmentioning

confidence: 99%

Bounds on the fault‐diameter of graphs

Dankelmann

2017

Networks

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Let G be a ( k + 1 ) ‐connected or ( k + 1 ) ‐edge‐connected graph, where k ∈ ℕ . The k‐fault‐diameter and k‐edge‐fault‐diameter of G is the largest diameter of the subgraphs obtained from G by removing up to k vertices and edges, respectively. In this paper we give upper bounds on the k‐fault‐diameter and k‐edge‐fault‐diameter of graphs in terms of order. We show that the k‐fault‐diameter of a ( k + 1 ) ‐connected graph G of order n is bounded from above by n − k + 1 , and by approximately 4 k + 2 n if G is also triangle‐free. If G does not contain 4‐cycles then this bound can be improved further to approximately 5 n ( k − 1 ) 2 . We further show that the k‐edge‐fault‐diameter of a ( k + 1 ) ‐edge‐connected graph of order n is bounded by n – 1 if k = 1, by ⌊ 2 n − 1 3 ⌋ if k = 2, and by approximately 3 k + 2 n if k ≥ 3 , and give improved bounds for triangle‐free graphs. Some of the latter bounds strengthen, in some sense, bounds by Erdös, Pach, Pollack, and Tuza (J Combin Theory B 47 (1989) 73–79) on the diameter. All bounds presented are sharp or at least close to being optimal. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 70(2), 132–140 2017

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Fault-tolerant diameter for three family interconnection networks

Shi

2010

J Comb Optim

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Fault diameter of product graphs

Cited by 7 publications

References 9 publications

Improved upper bounds for vertex and edge fault diameters of Cartesian graph bundles

Improved upper bounds for vertex and edge fault diameters of Cartesian graph bundles

Bounds on the fault‐diameter of graphs

Fault-tolerant diameter for three family interconnection networks

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