The complete bipartite graph $K_{4,4}$ is Uniformly Most-Reliable.

In network design, the all-terminal reliability maximization is of paramount importance. In this classical setting, we assume a simple graph with perfect nodes but independent edge failures with identical probability $\rho$. The goal is to communicate $n$ terminals using $e$ edges, in such a way that the connectedness probability of the resulting random graph is maximum. A graph with $n$ nodes and $e$ edges that meets the maximum reliability property for all $\rho \in (0,1)$ is called uniformly most-reliable $(n, e)$-graph (UMRG). The discovery of these graphs is a challenging problem that involves an interplay between extremal graph theory and computational optimization. Recent works confirm the existence of special cubic UMRGs, such as Wagner, Petersen and Yutsis graphs, and a 4-regular graph $H=\overline{C_7}$. In a foundational work in the field, Boesch. et. al. state with no proof that the bipartite complete graph $K_{4,4}$ is UMRG. In this paper, we revisit the breakthroughs in the theory of UMRG. A simple methodology to determine UMRGs based on counting trivial cuts is presented. Finally, we test this methodology to mathematically prove that the complete bipartite graph $K_{4,4}$ is UMRG.