The algorithm to insert an edge e in linear time into a planar graph G with a minimal number of crossings on e [14] is a helpful tool for designing heuristics that minimize edge crossings in topological drawings of general graphs. Unfortunately, not all such topological drawings are stretchable, i.e., there may not exist an equivalent straight-line drawing. That is, there is no planar straight-line drawing Γ of G such that in $\mathrm{\Gamma }+e$ the edge e crosses the same edges as in the topological drawing of $G+e$ and it does so in the same order. This motivates the study of the computational complexity of the problem Geometric Edge Insertion: Given a combinatorially embedded graph G, compute a geometric embedding Γ of G that minimizes the crossings in $\mathrm{\Gamma }+e$.

We give a characterization of the stretchable topological drawings of $G+e$ that also applies to the case where the outer face is fixed; this answers an open question of Eades et al. [8]. Algorithmically, we focus on the case where the outer face is not fixed. We show that Geometric Edge Insertion can be solved efficiently for graphs of maximum degree 5. For the general case, we show a $(\mathrm{\Delta }-2)$-approximation, where Δ is the maximum vertex degree of G and an FPT algorithm with respect to the minimum number of crossings. Finally, we consider the problem of testing whether there exists a solution of Geometric Edge Insertion that achieves the lower bound obtained by a topological insertion.

该算法以插入的边缘È线性时间为平面图形ģ与最小数量的交叉的ë [14]是用于设计启发式，最大限度地减少在一般图的拓扑附图边交叉的有用工具。不幸的是，并非所有这样的拓扑图都是可拉伸的，即，可能不存在等效的直线图。也就是说，不存在G 的平面直线图 Γ使得在$\mathrm{\Gamma }+\mathrm{电子}$边e与拓扑图中的相同边相交$G+\mathrm{电子}$它以相同的顺序这样做。这激发了问题的计算复杂度的研究几何边缘插入：给定一个组合地嵌入图表ģ，计算的几何嵌入Γ ģ最小化的交叉中$\mathrm{\Gamma }+\mathrm{电子}$.

我们给出了可拉伸拓扑图的表征 $G+\mathrm{电子}$这也适用于外表面固定的情况；这回答了 Eades 等人的一个悬而未决的问题。[8]。在算法上，我们专注于外表面不固定的情况。我们表明几何边插入可以有效地解决最大度数为 5 的图。对于一般情况，我们展示了$(\mathrm{\Delta }-2)$- 近似，其中 Δ 是G的最大顶点度数和 FPT 算法关于最小交叉次数。最后，我们考虑测试是否存在实现拓扑插入获得的下界的几何边插入的解决方案的问题。