Abstract Restricted edge connectivity is a more refined network reliability index than edge connectivity. For a connected graph G = (V, E), an edge set S ⊆ E is a restricted edge cut if G − S is disconnected and every component of G − S has at least two vertices. The restricted edge connectivity of G is defined as the cardinality of a minimum restricted edge cut. G is super restricted edge connected if every minimum restricted edge cut of G isolates one edge. In this article, we present several neighborhood conditions for a graph to be super restricted edge connected. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010

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. © 2007 Wiley Periodicals, Inc. NETWORKS, 2008