Real Reliability Roots of Simple Graphs are Dense

Abstract

We prove that the closure of the real roots of all-terminal reliability polynomials is exactly $[-1,0] \cup \{1\}$, resolving a conjecture of Brown and McMullin and refining the corresponding density result for multigraphs due to Brown and Colbourn. The crux of the proof is demonstrating that real reliability roots of edge-substitution graphs $G[H]$, where $G$ ranges over connected multigraphs and $H$ ranges over complete graphs missing an edge, are dense.