On graphs of invulnerable communication nets

Graph theoretic techniques provide a convenient tool for the investigation of the vulnerability of a communication network to either natural or enemy-induced damage. In this paper, the communication network is represented by a nonoriented linear graph, and the problem of synthesizing networks to provide optimal. protection against damage is investigated. Damage is associated with the removal of a set of nodes that disconnects the graph or by the removal of a set of edges that disconnects the graph. It is assumed that fixed cost for edges may be envisioned as prescribing the total number of edges allowed to connect a given number of nodes. A class of graphs that provides optimum damage protection for a fixed cost is then derived. Bipartite graphs are investigated in detail. It is shown that the complete bipartite graph is also an optimal damage-resistant net and that it can be decomposed into Hamiltonian cycles to yield optimal graphs of lower order. Another possible criterion of optimality is introduced and studied. This new measure is the minimum number of edges that must be deleted in order to isolate a given number of nodes. It is then shown that the complete bipartite graph is also optimal with respect to this alternative measure of vulnerability. This paper also contains proofs of several new and interesting graph theoretic results.