A node of a graph G, thought of as representing a communication network, is said to be redundant provided that its removal does not diminish the connectivity. In constructing networks, we require reliable connectedness in addition to the usual requirement of reliability (i.e., the higher the connectivity, the more reliable the network). Two nodes are called reliably connected if they are joined by a reliable path, i.e., a path whose internal nodes (if any) are all redundant. For a given n, we construct a communication network on n nodes having a given value k of connectivity (reliability condition) and as few edges as possible (optimization condition) and in which any pair of nodes, with one exceptional node, are reliably connected (reliable connectedness condition). We also studied the associated-with-G strong redundancy graph and a sequence of weak redundancy graphs, quotient graphs which display decompositions of G into connected subgraphs in accordance with the contributions of the nodes to the connectivity (G).
