# What do you call a directed graph in which reachablility is a symmetric relation?

Let $(N,E)$ a directed graph in which, if $a$ is reachable from $b$, then $b$ is also reachable from $a$. In other words, if $a$ and $b$ lie on a common path, then they also lie on a common cycle.

Clearly, this is the case for any undirected graph (the cycle formulation doesn't work then), but in general not for directed graphs.