Simultaneous Embeddings with Few Bends and Crossings
Vol. 23, no. 4, pp. 683-713, 2019. Regular paper.
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.
Submitted: December 2018.
Reviewed: April 2019.
Revised: May 2019.
Reviewed: July 2019.
Revised: July 2019.
Accepted: July 2019.
Final: August 2019.
Published: September 2019.
Communicated by Stephen G. Kobourov
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