The use of computer communication networks has been rapidly increasing in order to: 1) share expensive hardware & software resources, and 2) provide access to main systems from distant locations. The reliability & cost of these systems are important and are largely determined by network topology. Network topology consists of nodes and the links between nodes. The selection of optimal network topology is an NP-hard combinatorial problem so that the classical enumeration-based methods grow exponentially with network size. In this study, a heuristic search algorithm inspired by evolutionary methods is presented to solve the all-terminal network design problem when considering cost & reliability. The genetic algorithm heuristic is considerably enhanced over conventional implementations to improve effectiveness & efficiency. This general optimization approach is computationally efficient and highly effective on a large suite of test problems with search spaces up to 2.10(90).