Based on a new version of Hopcroft and Tarjan's planarity testing algorithm, we develop an O (mlogn)-time algorithm to find a maximal planar subgraph. Key words. algorithm, complexity, depth-first-search, embedding, planar graph, selection tree AMS(MOS) subject classifications. 68R10, 68Q35, 94C15 1. Introduction In [15], Wu defined the problem of planar graphs in terms of the following four subproblems: ################## 1 This work was partly supported by Thomson-CSF/DSE and by the National Science Foundation under grant CCR9002428. 2. Research at Princeton University partially supported by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center, grant NSF-STC88-09648, and the Office of Naval Research, contract N00014-87-K-0467. -- -- - 2 - P1. Decide whether a connected graph G is planar. P2. Find a minimal set of edges the removal of which will render the remaining part of G planar. P3. Gi...

. Given an undirected graph G, the maximal planar subgraph problem is to determine a planar subgraph H of G such that no edge of G-H can be added to H without destroying planarity. Polynomial algorithms have been obtained by Jakayumar, Thulasiraman and Swamy [6] and Wu [9]. O(mlogn) algorithms were previously given by Di Battista and Tamassia [3] and Cai, Han and Tarjan [2]. A recent O(m a a a a a a a a a a a a a a a a a a a a a a a a a (n)) algorithm was obtained by La Poute [7]. Our algorithm is based on a simple planarity test [5] developed by the author, which is a vertex addition algorithm based on a depth-first-search ordering. The planarity test [5] uses no complicated data structure and is conceptually simpler than Hopcroft and Tarjan's path addition and Lempel, Even and Cederbaum's vertex addition approaches. 1 1. Introduction Given an undirected graph, the planarity testing problem is to determine whether there exists a clockwise edge ordering around each vertex such that t...

Previously counting embeddings of planar graphs [5] used P-Q trees and was restricted to biconnected graphs. Although the P-Q tree approach is conceptually simple, its implementation is complicated. In this paper we solve this problem using DFS trees, which are easy to implement. We also give formulas that count the number of embeddings of general planar graphs (not necessarily connected or biconnected) in O (n) arithmetic steps, where n is the number of vertices of the input graph. Finally, our algorithm can be extended to generate all embeddings of a planar graph in linear time with respect to the output. Key words. graph, depth first search, embedding, planar graph, articulation point, connected component AMS(MOS) subject classifications. 68R10, 68Q35, 94C15 1. Introduction In [14], Wu stated four basic planar graph problems: 1. Decide whether a connected graph G is planar. 2. Find a minimal set of edges the removal of which will render the remaining part of G planar. ...

We are taking the view that crossings of adjacent edges are trivial, and easily got rid of.