Super restricted edge-connectivity of graphs with diameter 2

Abstract

For a connected graph G, an edge-cut S is called a restricted edge-cut if G - S contains no isolated vertices. And G is said to be super restricted edge-connected, for short super-lambda', if each minimum restricted edge-cut of G isolates an edge. Let V-delta denote the set of the minimum degree vertices of G. In this paper, for a super-lambda' graph G with diameter D >= 2 and minimum degree delta >= 4, we show that the induced subgraph G vertical bar V-delta vertical bar contains no complete graph K delta-1. Applying this property we characterize the super restricted edge connected graphs with diameter 2 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]. (C) 2012 Elsevier B.V. All rights reserved.