Simultaneous Embeddings with Vertices Mapping to Pre-Specified Points

TL;DR

This paper introduces a technique for embedding planar graphs with fixed vertex mappings, minimizing edge bends, and extends it to simultaneous embeddings of multiple graphs, achieving optimal bounds on the number of bends per edge.

Contribution

The paper presents a novel method for fixed vertex embeddings of planar graphs with bounded bends, and extends it to simultaneous embeddings of multiple graphs with optimal bend bounds.

Findings

Any planar graph can be drawn with at most 3n + O(1) bends per edge.

For random planar graphs, 2n + O(1) bends per edge suffice on average.

Simultaneous embeddings of k graphs can be achieved with O(n^{1-1/k}) bends per edge, matching lower bounds.

Abstract

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.