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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.

MSC:

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

Keywords:

bipartite graph
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References:

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