Sufficient conditions for a graph to be super restricted edge‐connected

Abstract

Restricted edge connectivity is a more refined network reliability index than edge connectivity. A restricted edge cut F of a connected graph G is an edge cut such that G −F has no isolated vertex. The restricted edge connectivity λ is the minimum cardinality over all restricted edge cuts. We call G λ -optimal if λ = ξ , where ξ is the minimum edge degree in G. Moreover, a λ -optimal graph G is called a super restricted edge-connected graph if every minimum restricted edge cut separates exactly one edge. Let D and g denote the diameter and girth of G, respectively. In this paper, we first present a necessary condition for non-super restricted edge-connected graphs with minimum degree δ ≥ 3 and D ≤ g − 2. Next, we prove that a connected graph with minimum degree δ ≥ 3 and D ≤ g − 3 is super restricted edge-connected. Finally, we give some sufficient conditions on the conditional diameter and the girth for super restricted edge-connected graphs. Keywords: restricted edge connectivity; super restricted edgeconnected graphs; diameter; girth TERMINOLOGY AND INTRODUCTIONA processor interconnection network or a communications network is conveniently modeled by an undirected graph G = (V , E), in which the vertex set V corresponds to processors or switching elements, and the edge set E corresponds to communication links. One fundamental consideration in the design of networks is reliability. When studying network reliability, one often considers a network model [9] whose vertices are perfectly reliable while edges may fail independently with the same probability ρ ∈ (0, 1). For subsets A and A of V , we denote by and ξ = ξ(G) denote the minimum degree and the minimum edge degree in G, respectively. It is well known that λ ≤ δ for a general graph. If λ = δ, then G is said to be maximally edge-connected. To minimize m λ , Bauer et al. [6,7] defined the super-λ graphs. A graph G is said to be super-λ if each of its minimum edge cuts isolates a vertex. That is, if F is a set of λ edges such that G − F is disconnected, then F is the set of edges incident with a certain vertex of G. If G is super-λ, then λ = δ. But the converse is not true. As a more refined index than edge connectivity, restricted edge connectivity was proposed by Esfahanian and Hakimi [9]. A set of edges F in a connected graph G is called a restricted edge cut if G − F is disconnected and contains no isolated vertex. If such an edge cut exists, then the restricted edge connectivity of G, denoted by λ = λ (G), is defined to be the minimum number of edges over all restricted edge cuts of G. A restricted edge cut F is called a λ -cut if |F| = λ (G). A connected graph G is called λ -connected if λ (G) exists. Esfahanian and Hakimi [9] showed that each connected graph G of order ν ≥ 4 except a star K 1,ν−1 is λ -connected and satisfies λ(G) ≤ λ (G) ≤ ξ(G). So a connected graph G must be λ -connected if δ ≥ 3. A graph G is called a λ -optimal graph if λ (G) = ξ(G). Moreover, G is super restricted edgeconnected, in short, super-λ , if every minimum restricted

Super-connected but not super edge-connected graphs

“…For more results on super-connected graphs or super edge-connected graphs, we refer the reader to [6,7,10,11,13,[15][16][17].…”

Optimization problems of the third edge-connectivity of graphs

“…(2) [11] If d(x) + d(y) n + 2 (n 4) for any pair of nonadjacent vertices x and y in G, then G is super-λ 2 except that n is even and G is an (n/2, 2)-barbell. Now we extend Theorem 5 to the following Theorem 6.…”

“…For a further study on the maximality and superiority of the second edge-connectivity and their application to network reliability comparison, the interested readers are referred to refs. [5][6][7][8][9][10][11][12].…”

On restricted edge-connectivity of replacement product graphs

“…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]). …”