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 theminimumcardinality over all restricted edge cuts.WecallG λ′-optimal if λ′ = ξ , where ξ is theminimum edgedegree inG. Moreover, a λ′-optimal graphG 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, Vol. 51(3), 200–209 2008