Simultaneous Embeddings with Few Bends and Crossings

TL;DR

This paper demonstrates that simultaneous embeddings of two planar graphs can be achieved with few bends and crossings, improving previous bounds and providing specific constructions for different graph classes.

Contribution

The paper introduces new bounds on the number of bends and crossings needed for simultaneous embeddings of planar graphs, improving upon prior results.

Findings

Trees require at most one bend per edge and four crossings per edge pair.

Planar graph and tree pairs require six bends per edge and eight crossings per edge pair.

Planar graph pairs require six bends per edge and sixteen crossings per edge pair.

Abstract

A simultaneous embedding with fixed edges (SEFE) of two planar graphs $R$ and $B$ is a pair of plane drawings of $R$ and $B$ that coincide when restricted to the common vertices and edges of $R$ and $B$. We show that whenever $R$ and $B$ admit a SEFE, they also admit a SEFE in which every edge is a polygonal curve with few bends and every pair of edges has few crossings. Specifically: (1) if $R$ and $B$ are trees then one bend per edge and four crossings per edge pair suffice (and one bend per edge is sometimes necessary), (2) if $R$ is a planar graph and $B$ is a tree then six bends per edge and eight crossings per edge pair suffice, and (3) if $R$ and $B$ are planar graphs then six bends per edge and sixteen crossings per edge pair suffice. Our results improve on a paper by Grilli et al. (GD'14), which proves that nine bends per edge suffice, and on a paper by Chan et al. (GD'14), which proves that twenty-four crossings per edge pair suffice.