The edge fault-diameter of Cartesian graph bundles

Banič, Iztok; Erveš, Rija; Erovnik, Janez

doi:10.1016/j.ejc.2008.09.004

Cited by 13 publications

(25 citation statements)

References 18 publications

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“…For example [9,13] consider Hamiltonian properties assuming a combination of vertex and edge faults. In our recent work on fault diameter of Cartesian graph products and bundles [3][4][5]2], analogous results were found for both fault diameter and edge fault diameter. However, the proofs for vertex and edge faults in [3][4][5]2] are independent, and our effort to see how results in one case may imply the others was not successful.…”

Section: Introductionsupporting

confidence: 65%

Edge, vertex and mixed fault diameters

Banič

Erveš

Žerovnik

2009

Advances in Applied Mathematics

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Let ▫${mathcal{D}}^E_q(G)$▫ denote the maximum diameter among all subgraphs obtained by deleting ▫$q$▫ edges of ▫$G$▫. Let ▫${mathcal{D}}^V_p(G)$▫ denote the maximum diameter among all subgraphs obtained by deleting ▫$p$▫ vertices of ▫$G$▫. We prove that ▫${mathcal{D}}^E_a(G) leqslant {mathcal{D}}^V_a(G) + 1$▫ a for all meaningful ▫$a$▫. We also define mixed fault diameter ▫${mathcal{D}}^M_{(p,q)}(G)$▫, where ▫$p$▫ vertices and ▫$q$▫ edges are deleted at the same time. We prove that for ▫$0 < l leqslant a$▫, ▫${mathcal{D}}^E_a(G) leqslant {mathcal{D}}^M_{(a-ell,ell)}(G) leqslant {mathcal{D}}^V_a(G) + 1$▫, and give some examples

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Section: Introductionsupporting

confidence: 65%

Edge, vertex and mixed fault diameters

Banič

Erveš

Žerovnik

2009

Advances in Applied Mathematics

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“…We show by examples that the bounds of the new result are tight. In addition, in some cases Theorem 4.6 also improves previously known results on vertex and edge fault diameters on these classes of Cartesian graph bundles [2,5].…”

Section: Introductionsupporting

confidence: 68%

“…An upper bound for the mixed fault diameter of Cartesian graph bundles is given in [14] that in some case also improves previously known results on vertex and edge fault diameters on these classes of Cartesian graph bundles [2,5]. However these results address only the number of faults given by the connectivity of the fibre (plus one vertex), while the connectivity of the graph bundle can be much higher when the connectivity of B is substantial.…”

Section: Introductionsupporting

confidence: 58%

“…For example, [16,24] consider Hamiltonian properties assuming a combination of vertex and edge faults. In recent work on fault diameter of Cartesian graph products and bundles [2,3,4,5], analogous results were found for both fault diameter and edge fault diameter. However, the proofs for vertex and edge faults are independent, and our effort to see how results in one case may imply the others was not successful.…”

Section: Introductionsupporting

confidence: 51%

“…A lot of work has been done on various aspects of network fault tolerance, see for example the survey [8] and the more recent papers [16,24,26]. In particular the fault diameter with faulty vertices, which was first studied in [18], and the edge fault diameter have been determined for many important networks recently [2,3,4,5,10,11,19,25]. Usually either only edge faults or only vertex faults are considered, while the case when both edges and vertices may be faulty is studied rarely.…”

Section: Introductionmentioning

confidence: 99%

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Mixed fault diameter of Cartesian graph bundles II

Erveš¹,

Žerovnik²

2015

Ars Math. Contemp.

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The mixed fault diameter D (p,q) (G) is the maximum diameter among all subgraphs obtained from graph G by deleting p vertices and q edges. A graph is (p, q)+connected if it remains connected after removal of any p vertices and any q edges. Let F be a connected graph with the diameter D(F ) > 1, and B be (p, q)+connected graph. Upper bounds for the mixed fault diameter of Cartesian graph bundle G with fibre F over the base graph B are given. We prove that if q > 0, then

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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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The edge fault-diameter of Cartesian graph bundles

Cited by 13 publications

References 18 publications

Edge, vertex and mixed fault diameters

Edge, vertex and mixed fault diameters

Mixed fault diameter of Cartesian graph bundles II

Bounds on the fault‐diameter of graphs

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