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 connec-tivity 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 mini-mum 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 min-imum 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 dia-meter and the girth for super restricted edge-connecte