Does the existing characterization yield a polynomial-time recognition algorithm

Determine whether the characterization of bipartite graphs that admit 2-layer fan-planar drawings given by Binucci et al. (2017) can be algorithmically exploited to derive a polynomial-time recognition algorithm for deciding 2-layer fan-planarity of general bipartite graphs.

Background

Prior work by Binucci et al. (2017) provided a characterization of bipartite graphs that have 2-layer fan-planar drawings and a linear-time recognition algorithm for the special case of biconnected bipartite graphs. While this characterization clarified structural aspects, it was not established whether it directly yields a general polynomial-time recognition algorithm.

The present paper resolves the overall recognition problem via a different approach, but it does not address whether the previous characterization itself suffices to produce a polynomial-time algorithm, leaving that specific methodological question unclear.