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

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