On a kind of restricted edge connectivity of graphs

Meng, Jixiang; Ji, Youhu

doi:10.1016/s0166-218x(00)00337-1

Cited by 85 publications

(23 citation statements)

References 6 publications

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“…The restricted edge-connectivity provides a more accurate measure of fault-tolerance of networks than the edge-connectivity (see [4,5]). Thus, determining the value of λ ′ for some special classes of graphs or characterizing λ ′ -optimal graphs have received considerable attention in the literature (see, for instance, [5,11,12,20,23,24,31,32,33]). …”

Section: Preliminariesmentioning

confidence: 99%

On restricted edge-connectivity of replacement product graphs

Hong

2016

Sci. China Math.

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This paper considers the edge-connectivity and the restricted edge-connectivity of replacement product graphs, gives some bounds on edge-connectivity and restricted edge-connectivity of replacement product graphs and determines the exact values for some special graphs. In particular, the authors further confirm that under certain conditions, the replacement product of two Cayley graphs is also a Cayley graph, and give a necessary and sufficient condition for such Cayley graphs to have maximum restricted edge-connectivity. Based on these results, the authors construct a Cayley graph with degree d whose restricted edge-connectivity is equal to d + s for given odd integer d and integer s with d 5 and 1 s d − 3, which answers a problem proposed ten years ago.

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Section: Preliminariesmentioning

confidence: 99%

On restricted edge-connectivity of replacement product graphs

Hong

2016

Sci. China Math.

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“…A λ -connected graph G is super-λ if and only if either G is non-λ 3 -connected, or G is λ 3 -connected and λ (G) < λ 3 (G). For more information on order-3 edgeconnectivity λ 3 (G), we may refer to [5,14,18]. We use K 2 p to denote any graph obtained by joining two copies of the complete graph K p (p ≥ 2) with 2p additional edges such that d(v) = p + 1 for all vertices v ∈ V (K 2 p ).…”

Section: Edge-cut S Of G Is Called An Order-3 Edge-cut If Each Componmentioning

confidence: 99%

Sufficient conditions for graphs to be λ′‐optimal and super‐λ′

Shang

Zhang

2007

Networks

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An edge-cut S of a connected graph G is called a restricted edge-cut if G − S contains no isolated vertices. The minimum cardinality of all restricted edge-cuts is called the restricted edge-connectivity λ (G) of G. A graph G is said to be λ -optimal if λ (G) = ξ(G), where ξ(G) is the minimum edge-degree of G. A graph is said to be super-λ if every minimum restricted edge-cut isolates an edge.In this paper, first, we improve and generalize the sufficient conditions for λ -optimality in arbitrary graphs, bipartite graphs, and graphs with diameter 2, which were given by Hellwig and Volkmann, and show using examples that our results are best possible. Second, we provide a simple proof with less restrictive conditions than in Hellwig and Volkmann's theorem that gives sufficient conditions for λ -optimality in bipartite graphs. We conclude by presenting sufficient conditions for arbitrary, bipartite, and triangle-free graphs, and for graphs with diameter 2, to be super-λ respectively, and demonstrate that these conditions are best possible.

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“…For λ 1 , it has been extensively studied, (see, for example, [4,9,11,15]), and it is termed the restricted edge-connectivity denoted by the notation λ . For κ h , Esfahanian [3] determined κ 1 (Q n ) = 2n − 2, and this was further generalized by Latifi et al [8] …”

Section: κ(G) ≤ λ(G) ≤ δ(G) Where δ(G) Is the Minimum Degree Of G Amentioning

confidence: 99%

Super Connectivity of Line Graphs and Digraphs

Lü

2006

Acta Mathematicae Applicatae Sinica, English Series

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The h-super connectivity κ h and the h-super edge-connectivity λ h are more refined network reliability indices than the connectivity and the edge-connectivity. This paper shows that for a connected balanced digraph D and its line digraph L, if D is optimally super edge-connected, then κ 1 (L) = 2λ 1 (D), and that for a connected graph G and its line graph L, if one of κ 1 (L) and λ 2 (G) exists, then κ 1 (L) = λ 2 (G). This paper determines that κ 1 (B(d, n)) is equal to 4d − 8 for n = 2 and d ≥ 4, and to 4d − 4 for n ≥ 3 and d ≥ 3, and that κ 1 (K(d, n)) is equal to 4d − 4 for d ≥ 2 and n ≥ 2 except K(2, 2). It then follows that B(d, n) and K(d, n) are both super connected for any d ≥ 2 and n ≥ 1.

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On a kind of restricted edge connectivity of graphs

Cited by 85 publications

References 6 publications

On restricted edge-connectivity of replacement product graphs

On restricted edge-connectivity of replacement product graphs

Sufficient conditions for graphs to be λ′‐optimal and super‐λ′

Super Connectivity of Line Graphs and Digraphs

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