Simultaneous Embeddings with Few Bends and Crossings

Abstract

A simultaneous embedding with fixed edges ($\rm{S{\small EFE}}$) 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 $\rm{S{\small EFE}}$, they also admit a $\rm{S{\small EFE}}$ 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 simultaneously 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. (JGAA '15), which proves that twenty-four crossings per edge pair suffice.