On O(p2) algorithms for planarization

A recent paper of R. Jayakumar et al. (ibid., vol.8, p.257-67, 1989) presented, for a given 2-connected nonplanar graph G, two O(p²) planarization algorithms (where p is the number of vertices of G). The first algorithm provided a spanning planar subgraph H of G. Provided H is 2-connected, the second algorithm augmented H to a maximal planar subgraph H´ of G. Both algorithms used the PQ-tree data structure and a vertex addition approach (i.e., building the subgraph H by adding one vertex at a time). Here, it is shown that the vertex addition approach cannot guarantee the construction of a 2-connected maximal planar subgraph, even if one exists. Also, as a partial answer to an open question by Jayakumar et al. It is shown that it is not guaranteed that a maximal planar subgraph of a 2-connected graph is itself 2-connected, even under the more stringent requirement that the initial graph have minimum degree 3. Since the vertex addition approach using PQ-trees requires the graph to be 2-connected, this result indicates that the vertex addition approach may not be appropriate for the efficient solution of the maximal planarization problem