Sufficient Conditions for a Graph to be Super Restricted Edge-Connected

摘要

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.