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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
bipartite graph
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