Inserting an Edge into a Geometric Embedding

The algorithm of Gutwenger et al. to insert an edge $e$ in linear time into a planar graph $G$ with a minimal number of crossings on $e$, 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 $Γ$ such that $Γ+e$ 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 $Γ$ that has the same combinatorial embedding as $G$ and that minimizes the crossings of $Γ+e$. 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 $(Δ-2)$, where $Δ$ is the maximum vertex degree of $G$.