Uniformly Most Reliable Three-Terminal Graph of Dense Graphs

Abstract

A graph G with k specified target vertices in vertex set is a k -terminal graph. The k -terminal reliability is the connection probability of the fixed k target vertices in a k -terminal graph when every edge of this graph survives independently with probability p . For the class of two-terminal graphs with a large number of edges, Betrand, Goff, Graves, and Sun constructed a locally most reliable two-terminal graph for p close to 1 and illustrated by a counterexample that this locally most reliable graph is not the uniformly most reliable two-terminal graph. At the same time, they also determined that there is a uniformly most reliable two-terminal graph in the class obtained by deleting an edge from the complete graph with two target vertices. This article focuses on the uniformly most reliable three-terminal graph of dense graphs with n vertices and m edges. First, we give the locally most reliable three-terminal graphs of n and m in certain ranges for p close to 0 and 1. Then, it is proved that there is no uniformly most reliable three-terminal graph with specific n and m , where n ≥ 7 and n 2 − ⌊ n − 3 / 2 ⌋ ≤ m ≤ n 2 − 2 . Finally, some uniformly most reliable graphs are given for n vertices and m edges, where 4 ≤ n ≤ 6 and m = n 2 − 2 or n ≥ 5 and m = n 2 − 1 .