Simultaneous Embeddings with Vertices Mapping to Pre-Specified Points

We discuss the problem of embedding graphs in the plane with restrictions on the vertex mapping. In particular, we introduce a technique for drawing planar graphs with a fixed vertex mapping that bounds the number of times edges bend. An immediate consequence of this technique is that any planar graph can be drawn with a fixed vertex mapping so that edges map to piecewise linear curves with at most $3n + O(1)$ bends each. By considering uniformly random planar graphs, we show that $2n + O(1)$ bends per edge is sufficient on average. To further utilize our technique, we consider simultaneous embeddings of $k$ uniformly random planar graphs with vertices mapping to a fixed, common point set. We explain how to achieve such a drawing so that edges map to piecewise linear curves with $O(n^{1-1/k})$ bends each, which holds with overwhelming probability. This result improves upon the previously best known result of O(n) bends per edge for the case where $k \geq 2$. Moreover, we give a lower bound on the number of bends that matches our upper bound, proving our results are optimal.