It is known that a graph is 2-vertex-connected iff it has an (open) ear decomposition, and there is a linear-time algorithm for finding an ear decomposition.
I think it is also true that if a graph is 2-vertex-connected, we can find an ear decomposition starting with any arbitrary cycle as the first ear. Also, there should still be a linear-time algorithm for finding such an ear decomposition. However, I cannot find any reference for this. Is it stated somewhere?