A digraph(=directed graph = graph with directed/oriented edges) $X$ is said to be strongly connected if for any distinct vertices $v,w$, there is a directed path from $v$ to $w$.
In particular in such digraphs, every vertex has an incoming edge and an outgoing edge. My question is, whether the last property characterizes strongly connected digraphs? To be precise,
Q. Let $X$ be a connected, finite, digraph. Assume that every vertex has at least one incoming and one outgoing edge. Is $X$ strongly connected?