Simultaneous FPQ-ordering and hybrid planarity testing

We introduce and study 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 novel definition of fixedness that makes it possible to simplify and extend known techniques about SIMULTANEOUS PQ-ORDERING. We exploit this result to study different versions of the hybrid planarity testing problem. Namely, we show polynomial-time solutions for a variant of NODETRIX PLANARITY with fixed sides, for POLYLINK PLANARITY, and for CLIQUE PLANARITY with fixed sides. (C) 2021 Elsevier B.V. All rights reserved.