##
**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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XMLCite

\textit{P. Eades} and \textit{S. Whitesides}, Theor. Comput. Sci. 131, No. 2, 361--374 (1994; Zbl 0819.68086)

Full Text:
DOI

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