On the complexity of the maximum biplanar subgraph problem

Let G ˆ …S; T ;E† be a bipartite graph with vertex set S [ T and edge E S T . A typical convention for drawing G is to put the vertices of S on a line and the vertices of T on a separate parallel line and then represent edges by placing straight line segments between the vertices that determine them. In this convention, a drawing is biplanar if edges do not cross, and a subgraph of G is biplanar if it has a biplanar drawing. The maximum biplanar subgraph problem is to ®nd a biplanar subgraph with a maximum number of edges. In general, this maximum biplanar subgraph problem is NP-complete. In this paper, we show the maximum biplanar subgraph problem belongs not only to the class P, but also the class NC, when input graphs are restricted to doubly convexbipartite graphs which is an important subclass of bipartite graphs. Moreover, our sequential algorithm is optimal. Ó 2000 Elsevier Science Inc. All rights reserved.