On the -optimality in graphs with odd girth and even girth

For a connected graph G , the restricted edge-connectivity λ ′ ( G ) is defined as the minimum cardinality of an edge-cut over all edge-cuts S such that there are no isolated vertices in G − S . A graph G is said to be λ ′ -optimal if λ ′ ( G ) = ξ ( G ) , where ξ ( G ) is the minimum edge-degree in G defined as ξ ( G ) = min { d ( u ) + d ( v ) − 2 : u v ∈ E ( G ) } , d ( u ) denoting the degree of a vertex u . The main result of this paper is that graphs with odd girth g and finite even girth h ≥ g + 3 of diameter at most h − 4 are λ ′ -optimal. As a consequence polarity graphs are shown to be λ ′ -optimal.