It is known that a graph is 2-vertex-connected iff it has an [(open) ear decomposition](https://en.wikipedia.org/wiki/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?



Ear decomposition with initial cycle
The [paper of Schmidt](https://arxiv.org/ftp/arxiv/papers/1209/1209.0700.pdf) that you found answers the question in the positive, both existence and algorithm. Start the DFS on $C$ and preferentially follow edges of $C$ until the final edge, which becomes a back-edge to the root. Then in the second stage you can choose $C\_1=C$.


