On locally most reliable three-terminal graphs of sparse graphs

Abstract

A network structure with $ n $ vertices and $ m $ edges is practically represented by a graph with $ n $ vertices and $ m $ edges. The graph with $ k $ fixed target vertices is called a $ k $-terminal graph. This article studies the locally most reliable simple sparse three-terminal graphs, in which each edge survives independently with probability $ p $. For $ p $ close to $ 0 $ or $ 1 $, the locally most reliable three-terminal graphs with $ n $ vertices and $ m $ edges are determined, where $ n\geq5 $ and $ 9\leq m\leq4n-10 $. Finally, we prove that there is no uniformly most reliable three-terminal graph for $ n\geq5 $, $ 11