Drawing Partially Embedded and Simultaneously Planar Graphs

TL;DR

This paper presents algorithms for constructing planar graph drawings with few bends and crossings, extending existing results to partially embedded and simultaneous planarity problems, with efficient solutions given fixed embeddings.

Contribution

It generalizes classic planar drawing results by providing linear-bend and crossing bounds for partially embedded and simultaneous planarity problems with efficient algorithms.

Findings

Existence of planar drawings with linear bends per edge when embeddings are fixed.

Planar drawings with a constant number of crossings between edges in simultaneous planarity.

Efficient algorithms for constructing such drawings given the combinatorial embedding.

Abstract

We investigate the problem of constructing planar drawings with few bends for two related problems, the partially embedded graph problem---to extend a straight-line planar drawing of a subgraph to a planar drawing of the whole graph---and the simultaneous planarity problem---to find planar drawings of two graphs that coincide on shared vertices and edges. In both cases we show that if the required planar drawings exist, then there are planar drawings with a linear number of bends per edge and, in the case of simultaneous planarity, a constant number of crossings between every pair of edges. Our proofs provide efficient algorithms if the combinatorial embedding of the drawing is given. Our result on partially embedded graph drawing generalizes a classic result by Pach and Wenger which shows that any planar graph can be drawn with a linear number of bends per edge if the location of each vertex is fixed.