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Let $G=(V,E)$ be a simple graph with $|V|=n$ nodes and $|E|=m$ links, asubset $K \subseteq V$ of \emph{terminals}, a vector $p=(p_1,\ldots,p_m) \in[0,1]^m$ and a positive integer $d$, called \emph{diameter}. We assume nodesare perfect but links fail stochastically and independently, with probabilities$q_i=1-p_i$. The \emph{diameter-constrained reliability} (DCR for short), isthe probability that the terminals of the resulting subgraph remain connectedby paths composed by $d$ links, or less. This number is denoted by$R_{K,G}^{d}(p)$.  The general DCR computation is inside the class of$\mathcal{N}\mathcal{P}$-Hard problems, since is subsumes the complexity that arandom graph is connected. In this paper, the computational complexity ofDCR-subproblems is discussed in terms of the number of terminal nodes $k=|K|$and diameter $d$. Either when $d=1$ or when $d=2$ and $k$ is fixed, the DCR isinside the class $\mathcal{P}$ of polynomial-time problems. The DCR turns$\mathcal{N}\mathcal{P}$-Hard when $k \geq 2$ is a fixed input parameter and$d\geq 3$.  The case where $k=n$ and $d \geq 2$ is fixed are not studied in priorliterature. Here, the $\mathcal{N}\mathcal{P}$-Hardness of this case isestablished.

Diameter Constrained Reliability: Computational Complexity in terms of the diameter and number of terminals