Edge Fault-Diameter of Cartesian Product of Graphs

Banič, Iztok; Žerovnik, Janez

doi:10.1007/978-3-540-72951-8_19

Cited by 6 publications

(10 citation statements)

References 13 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%

“…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. A natural question remains whether it is possible to design a uniform theory that would enable unified proofs or provide tools to translate results for one type of faults to the other.…”

Section: Introductionsupporting

confidence: 65%

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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%

Section: Introductionsupporting

confidence: 65%

Edge, vertex and mixed fault diameters

Banič

Erveš

Žerovnik

2009

Advances in Applied Mathematics

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“…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: 64%

“…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%

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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“…A lot of work has been done on various aspects of network fault tolerance, see for example the survey [9] and the more recent papers [18,26,29]. In particular the fault diameter with faulty vertices, which was first studied in [20], and the edge fault diameter have been determined for many important networks recently [2,3,4,5,11,12,21,27]. 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: Motivation -Interconnection Networkmentioning

confidence: 99%

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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Edge Fault-Diameter of Cartesian Product of Graphs

Cited by 6 publications

References 13 publications

Edge, vertex and mixed fault diameters

Edge, vertex and mixed fault diameters

Mixed fault diameter of Cartesian graph bundles II

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

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