  We study the interplay between embedding constrained planarity and hybrid
planarity testing. We consider a constrained planarity testing problem, called
1-Fixed Constrained Planarity, and prove that this problem can be solved in
quadratic time for biconnected graphs. Our solution is based on a new
definition of fixedness that makes it possible to simplify and extend known
techniques about Simultaneous PQ-Ordering. We apply these results to different
variants of hybrid planarity testing, including a relaxation of NodeTrix
Planarity with fixed sides, that allows rows and columns to be independently
permuted.
