Directed walks, trails, paths and cycles, and strong connectivity

Let $D=(V,A)$ be a finite digraph (Multigraphs, loops and directed graphs as variants distinct from the default finite simple graph). A directed walk of length $\mathrm{\ell }$ is a vertex list $v_0,v_1,\dots ,v_{\mathrm{\ell }}$ such that $(v_{i-1},v_i)\in A$ for every $1\le i\le \mathrm{\ell }$. It is closed when $v_0=v_{\mathrm{\ell }}$.

A directed trail is a directed walk whose traversed arcs are distinct. A directed path is a directed walk whose vertices are distinct. A directed cycle is a closed directed walk $v_0,\dots ,v_{\mathrm{\ell }-1},v_{\mathrm{\ell }}=v_0$ of length $\mathrm{\ell }\ge 3$ in which $v_0,\dots ,v_{\mathrm{\ell }-1}$ are distinct. The lower bound agrees with the library's undirected cycle convention in Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges.

A digraph is strongly connected when its vertex set is nonempty and for every ordered pair $u,v$ of vertices there is a directed path from $u$ to $v$. It is weakly connected as defined through its underlying graph in Degree in a multigraph, indegree and outdegree in a digraph, and their underlying connectivity.