Let be a simple graph with vertices and edges, a subset of terminals, a vector and a positive integer , called the diameter. We assume vertices are perfect but edges fail stochastically and independently, with probabilities . The diameter constrained reliability (DCR) is the probability that the terminals of the resulting subgraph remain connected by paths composed of or fewer edges. This number is denoted by . The general DCR computation problem belongs to the class of ‐hard problems. The contributions of this article are threefold. First, the computational complexity of DCR‐subproblems is discussed in terms of the number of terminal vertices and the diameter . Either when or when and is fixed, the DCR problem belongs to the class of polynomial‐time solvable problems. The DCR problem becomes ‐hard when is a fixed input parameter and . The cases where or is a free input parameter and is fixed have not been studied in the prior literature. Here, the ‐hardness of both cases is established. Second, we categorize certain classes of graphs that allow the DCR computation to be performed in polynomial time. We include graphs with bounded corank, graphs with bounded genus, planar graphs, and in particular, Monma graphs, which are relevant in robust network design. Third, we introduce the problem of analyzing the asymptotic properties of the DCR measure in networks that grow infinitely following given probabilistic rules. We introduce basic results for Gilbert's random graph model. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 66(4), 296–305 2015