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...

The problem of finding a maximum spanning planar subgraph of a nonplanar graph is NP-Complete. Several heuristics for the problem have been devised but their worst-case performance is unknown, although a trivial lower bound of 1/3 the optimum number of edges is easily shown. We discuss a new heuristic, based on spanning trees, for generating a subgraph with size at least 2/3 of the optimum for any input graph. The skewness of the n-dimensional hypercube Qn is also derived. Finally, we explore the relationship between the skewness and crossing number of a graph.