My SODA talk slides on finding laminar 3-vertex separators of planar graphs are online: http://www.ics.uci.edu/~eppstein/pubs/EppRee-SODA-19-slides.pdf

At the talk, someone asked about finding a maximum set of laminar 3-separators. It's NP-complete: use Uehara's "NP-complete problems on a 3-connected cubic graph and their applications" (https://www.jaist.ac.jp/~uehara/pdf/triangle.pdf) to find cubic planar graphs whose ≤3-separators are vertex neighborhoods (so laminar separators = neighbors of independent vertices) in which maximum independent set is hard.