Abstract
The algorithm to insert an edge e in linear time into a planar graph G with a minimal number of crossings on e [10], is a helpful tool for designing heuristics that minimize edge crossings in drawings of general graphs. Unfortunately, some graphs do not have a geometric embedding \(\varGamma \) such that \(\varGamma +e\) has the same number of crossings as the embedding \(G+e\). This motivates the study of the computational complexity of the following problem: Given a combinatorially embedded graph G, compute a geometric embedding \(\varGamma \) that has the same combinatorial embedding as G and that minimizes the crossings of \(\varGamma +e\). We give polynomial-time algorithms for special cases and prove that the general problem is fixed-parameter tractable in the number of crossings. Moreover, we show how to approximate the number of crossings by a factor \((\varDelta -2)\), where \(\varDelta \) is the maximum vertex degree of G.
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