Graphs with large restrained domination number

Let G=(V,E) be a graph. A set S⊆V is a restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in V - S. The restrained domination number of G, denoted by γ_r(G), is the minimum cardinality of a restrained dominating set of G. Domke et al., submitted [3] showed that if a connected graph G of order n has minimum degree at least 2 and is not one of eight exceptional graphs, then γ_r(G)≤(n-1)/2. In this paper, we characterise those graphs of order n which are edge-minimal with respect to satisfying G connected, δ(G)≥2 and γ_r(G)≥(n-1)/2.