In a directed graph $G$, what do we call a path, a sequence of edges $$(v_0,v_1),(v_1,v_2),\dots,(v_{n-1},v_n)$$ of length $n$, that intersects every other path of the same length $$(w_0,w_1),(w_1,w_2),\dots,(w_{n-1},w_n)$$ in the sense that for each such $w$ there is a $k$ with $v_k=w_k$?
For instance, in a graph with two vertices $q$ and $r$ and only the three edges $(q,r),(r,r),(r,q)$, then $$(q,r),(r,r),(r,q)$$ is such a path, since any counterexample would start $$(r,q),(q,q)$$ ... but $(q,q)$ is not an edge.
