Diameter constrained reliability: Complexity and distinguished topologies

Let G = (V,E) be a simple graph with |V| = n nodes and |E| = m links, a subset K ⊆ V of terminals, a vector p = (p₁, ..., p_m) ∈ [0, 1]^m and a positive integer d, called diameter. We assume nodes are perfect but links fail stochastically and independently, with probabilities q_i = 1 - p_i. The diameter-constrained reliability (DCR for short), is the probability that the terminals of the resulting subgraph remain connected by paths composed by d links, or less. This number is denoted by R_K,G^d(p).