Super restricted edge-connectivity of graphs with diameter 2

For a connected graph GG, an edge-cut SS is called a restricted edge-cut if G−SG−S contains no isolated vertices. And GG is said to be super restricted edge-connected, for short super-λ′λ′, if each minimum restricted edge-cut of GG isolates an edge. Let VδVδ denote the set of the minimum degree vertices of GG. In this paper, for a super-λ′λ′ graph GG with diameter D≥2D≥2 and minimum degree δ≥4δ≥4, we show that the induced subgraph G[Vδ]G[Vδ] contains no complete graph Kδ−1Kδ−1. Applying this property we characterize the super restricted edge connected graphs with diameter 22 which satisfy a type of neighborhood condition. This result improves the previous related one which was given by Wang et al. [S. Wang, J. Li, L. Wu, S. Lin, Neighborhood conditions for graphs to be super restricted edge connected, Networks 56 (2010) 11–19].