Crossing properties of graph reliability functions

Abstract

Let G be a graph and p ϵ (0, 1). Let A(G, p) denote the probability that if each edge of G is selected at random with probability p then the resulting spanning subgraph of G is connected. Then A(G, p) is a polynomial in p. We prove that for every integer k ≥ 1 and every k-tuple (m1, m2, … ,mk) of positive integers there exist infinitely many pairs of graphs G1 and G2 of the same size such that the polynomial A(G1, p) − A(G2, p) has exactly k roots x1 < x2 < ··· < xk in (0, 1) such that the multiplicity of xi is mi. We also prove the same result for the two-terminal reliability polynomial, defined as the probability that the random subgraph as above includes a path connecting two specified vertices. These results are based on so-called A- and T-multiplying constructions that are interesting in themselves. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 206–221, 2000