Distinguished vertices in probabilistic rooted graphs

The expected number of vertices that remain joined to the root vertex s of a rooted graph Gs when edges are prone to fail is a polynomial EV (Gs; 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 (Gs; p), proving that the number of vertices of G is bounded by the size of the largest rational zero.