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Drawing graphs in two layers. (English) Zbl 0819.68086
Summary: Let \(G= (U, L, E)\) be a bipartite graph with vertex set \(U\cup L\) and edge set \(E\subseteq U\times L\). A typical convention for drawing \(G\) is to put the vertices of \(U\) on the 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 position for the vertices in either \(U\) or \(L\) are specified; (3) when the position 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.

68R10 Graph theory (including graph drawing) in computer science
68Q25 Analysis of algorithms and problem complexity
05C10 Planar graphs; geometric and topological aspects of graph theory
bipartite graph
Full Text: DOI
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