On Optimal 2 and 3Planar Graphs
Abstract
A graph is $k$planar if it can be drawn in the plane such that no edge is crossed more than $k$ times. While for $k=1$, optimal $1$planar graphs, i.e., those with $n$ vertices and exactly $4n8$ edges, have been completely characterized, this has not been the case for $k \geq 2$. For $k=2,3$ and $4$, upper bounds on the edge density have been developed for the case of simple graphs by Pach and Tóth, Pach et al. and Ackerman, which have been used to improve the wellknown "Crossing Lemma". Recently, we proved that these bounds also apply to nonsimple $2$ and $3$planar graphs without homotopic parallel edges and selfloops. In this paper, we completely characterize optimal $2$ and $3$planar graphs, i.e., those that achieve the aforementioned upper bounds. We prove that they have a remarkably simple regular structure, although they might be nonsimple. The new characterization allows us to develop notable insights concerning new inclusion relationships with other graph classes.
 Publication:

arXiv eprints
 Pub Date:
 March 2017
 arXiv:
 arXiv:1703.06526
 Bibcode:
 2017arXiv170306526B
 Keywords:

 Computer Science  Computational Geometry;
 Computer Science  Discrete Mathematics