Abstract
Computing a crossing minimum drawing of a given planar graph G augmented by an additional edge e where all crossings involve e, has been a long standing open problem in graph drawing. Alternatively, the problem can be stated as finding a combinatorial embedding of a planar graph G where the given edge e can be inserted with the minimum number of crossings. Many problems concerned with the optimization over the set of all combinatorial embeddings of a planar graph turned out to be NP-hard. Surprisingly, we found a conceptually simple linear time algorithm based on SPQR-trees, that is able to find a solution with the minimum number of crossings.
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Gutwenger, C., Mutzel, P. & Weiskircher, R. Inserting an Edge into a Planar Graph. Algorithmica 41, 289–308 (2005). https://doi.org/10.1007/s00453-004-1128-8
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DOI: https://doi.org/10.1007/s00453-004-1128-8