Given any asymmetric relationship $ A subseteq V ^ 2 $ a digraph $ D = (V, A) $ is minimally strong *iff* $ D $ is strongly connected and for each arc $ a in A $ the digraph $ D – a = (V, A setminus {a }) $ not strongly connected

Now for Robbins' theorem every $ 2 $The connected edge chart has a strong orientation, so clearly every biconnected chart must have a strong orientation. However, not all graphics have a minimally strong orientation, for example, wheel graphics https://en.wikipedia.org/wiki/Wheel_graph can never have a minimally strong orientation. So which biconnected graphics have a minimally strong orientation?

I can prove that if there is a biconnected graph $ G $ has a minimally strong orientation then $ chi (G) leq 3 $ as well as that $ | V (G) | leq | E (G) | leq 2 | V (G) | -3 $ also if we contract any triangle $ T subseteq G $ to form a graph $ G / T $ then this chart should also have a minimally strong orientation, similarly consider if $ G $ it has a minimally strong orientation, then each subdivision of $ G $. It is also quite simple to prove that each graph that does not have cycle chords must have a minimally strong orientation. Also the chart $ G $ It will always have a vertex with a degree equal to two.

However, I am not sure how to proceed from here, at the bottom of the second page of this document, the author briefly mentions these graphics and the article itself takes time to derive a technique to build minimally strong digraphs, although it is a Little hard to follow. , it seems that you are using a combination of clusters of two vertices and subdivisions in directed graphs defined recursively. Therefore, I believe that by studying the SPQR tree of these graphs, one might be able to derive some simple criteria for which the connected graphs have a minimally strong orientation. Although I am not familiar with all this and I wonder if this is excessive. Can anyone help me here?