Maximal Intervals of Decrease and Inflection Points for Node Reliability

Abstract

The \textit{node reliability} of a graph $G$ is the probability that at least one node is operational and that the operational nodes can all communicate in the subgraph that they induce, given that the edges are perfectly reliable but each node operates independently with probability $p\in[0,1]$. We show that unlike many other notions of graph reliability, the number of maximal intervals of decrease in $[0,1]$ is unbounded, and that there can be arbitrarily many inflection points in the interval as well.