Let $G=(V,E)$ be a directed graph with $n$ vertices and $m$ edges. The graph $G$ is called singly-connected if for each pair of vertices $v,w \in V$ there is at most one simple path from $v$ to $w$ in $G$. Buchsbaum and Carlisle (1993) gave an algorithm for testing whether $G$ is singly-connected in $O(n^{2})$ time. In this paper we describe a refined version of this algorithm with running time $O(s\cdot t+m)$, where $s$ and $t$ are the number of sources and sinks, respectively, in the reduced graph $G^{r}$ obtained by first contracting each strongly connected component of $G$ into one vertex and then eliminating vertices of indegree or outdegree $1$ by a contraction operation. Moreover, we show that the problem of finding a minimum cardinality edge subset $C\subseteq E$ (respectively, vertex subset $F\subseteq V$) whose removal from $G$ leaves a singly-connected graph is NP-hard.