Reliability Polynomials of Networks with Vertex Failure

For a graph G with perfectly reliable edges and unreliable vertices, we consider the reliability of G for which vertices fail independently of each other with a constant probability p. The reliability of graph G, denoted by P_n(G, p), is defined to be the probability that the induced subgraphs of surviving vertices connected. In this paper, our main goal is to investigate the reliability polynomials P_n(G, p) of G. The subgraph expression of reliability polynomial of a graph G are given.