Finding uniformly most reliable graphs by counting trivial cuts

TL;DR

This paper introduces a new methodology for identifying uniformly most reliable graphs by analyzing trivial cuts, successfully proving that certain bipartite complete graphs are UMRGs, thus advancing reliability optimization in network theory.

Contribution

The paper presents a novel approach based on bounding trivial cuts to determine UMRGs, providing the first mathematical proof for specific bipartite complete graphs.

Findings

Proves that K_{3,3} and K_{4,4} are UMRGs

Introduces a methodology based on trivial cuts for reliability analysis

Advances understanding of network reliability optimization

Abstract

There is a vast literature focused on network reliability evaluation. In the last decades, reliability optimization has been also addressed. Frank Boesch in 1986 introduced the concept of uniformly most reliable graph (UMRG). Later, Boesch \emph{et al.} presented the first UMRGs and conjectured that some special subdivisions of the bipartite complete graph $K_{3,3}$, as well as the bipartite complete graph $K_{4,4}$, are UMRGs. Wang proved that the first conjecture is true. Wendy Myrvold confirmed that $K_{4,4}$ is also UMRG, by means of computational tests. However, thus far, there is no mathematical proof in the literature. A trivial cut is an edge-set that includes all the incident edges of a fixed node. In this article we describe a methodology to determine UMRGs based on bounding the number of trivial cuts. As a proof-of-concept it is proved that both $K_{3,3}$ and $K_{4,4}$ are UMRGs.