On the λ′λ′-optimality in graphs with odd girth gg and even girth hh

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