Drawing Graphs in Two Layers

Abstract Let G=(U,L,E) be a bipartite graph with vertex set U ∪ L and edge set E ⊆ U x L. A typical convention for drawing G is to put the vertices of U on a line and the vertices of L on a separate, parallel line and then to represent edges by placing open 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 main results of this paper are the following: (1) it is NP-complete to determine whether G has a biplanar subgraph with at least K edges; (2) it is also NP-complete to determine whether G has such a subgraph when the positions for the vertices in either U or L are specified; (3) when the positions of the vertices in both U and L are specified, the problem can be solved in polynomial time by transformation to the longest ascending subsequence problem.

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