Distinguished vertices in probabilistic rooted graphs

Abstract

The expected number of vertices that remain joined to the root vertex s of a rooted graph G s when edges are prone to fail is a polynomial EV(G s ; p) in the edge probability p that depends on the location of s. We show that optimal locations for the root can vary arbitrarily as p varies from 0 to 1 by constructing a graph in which every permutation of k -specified vertices is the "optimal" ordering for some p, 0 < p < 1. We also investigate zeroes of EV(G s ; p), proving that the number of vertices of G is bounded by the size of the largest rational zero.