Optimal assignments for consecutive-2 graphs

Let G represent the graph structure of a system of components each of which can either work or fail. Suppose that the system itself fails if and only if two adjacent vertices both fail; then G is called a consecutive-2 graph. Given a set of probabilities $p = \{ p_1 , \cdots ,p_n \} $ where n is the number of vertices in G, the problem is to assign $p_i $ to the n vertices to minimize the probability of the system failing. Previous literature has dealt with the case that G consists of lines and cycles. Here we give some results applicable to general graphs. We also discuss the conditions for G to have an optimal assignment which depends only on the ranks of $p_i $.