Uniformly Optimally Reliable Graphs: A Survey

Abstract

Which is the most reliable graph with n nodes and m edges? This celebrated problem has several aspects, according to the notion of optimality (in a local or uniform sense), failure type (either nodes or edges), or reliability model (all-terminal connectedness, two-terminal or multiterminal setting). This article presents a chronological survey of the multiple proposals to address the problem, together with recent trends and enigmatic conjectures posed decades ago that promote further research.