O(n2) algorithms for graph planarization

The authors present two O(n²) planarization algorithms, PLANARIZE and MAXIMAL-PLANARIZE. These algorithms are based on A. Lempel, S. Even, and I. Cederbaum´s (1967) planarity testing algorithm and its implementation using PQ-trees. Algorithm PLANARIZE is for the construction of a spanning planar subgraph of an n-vertex nonplanar graph. The algorithm proceeds by embedding one vertex at a time and, at each step, adds the maximum number of edges possible without creating nonplanarity of the resultant graph. Given a biconnected spanning planar subgraph G_p of a nonplanar graph G, the MAXIMAL-PLANARIZE algorithm constructs a maximal planar subgraph of G which contains G_p . This latter algorithm can also be used to planarize maximally a biconnected planar graph