Very strongly connected graphs

A simple connected graph is strongly connected if there exists a directed path between every pair of vertices in both directions. Robbins showed that every \(2\)-edge-connected graph can be given edge orientations that result in the graph being strongly connected. But a random assignment of edge directions may or may not result in a graph being strongly connected. We say that a \(2\)-edge-connected graph is very strongly connected if any choice of edge orientations that does not feature a vertex having maximal or minimal indegree yields a strongly connected graph. We classify all graphs as either very strongly connected or not very strongly connected.