Let G be a simple graph with e perfectly reliable edges and n nodes which fail independently and with the same probability ρ. The residual connectedness reliability R(G,ρ) of G is the probability that the graph induced by the surviving nodes is connected. If ¦)(n,e) is the collection of all n node e edge simple graphs, then G∈¦)(n,e) is uniformly most reliable if R(G,ρ)∈R(G',ρ) for all G'∈¦)(n,e) and all 0∈ρ∈1. If S3(G) is the number of three point induced connected subgraphs of G, then G∈¦)(n,e) is S3-maximum if S3(G)∈S3(G') for all G'∈¦)(n,e). it is known that if G∈¦)(n,e) is S3-maximum and ρ is sufficiently large then R(G,ρ)∈R(G,ρ') for all non-S3-maximum graphs G'∈¦)(n,e). This paper characterizes the S3-maximum graphs in ¦)(n,e) for the range e∈(n2/4)+(2n-3)/4.
