2024/08/26 by Chaplick, Steven, Förster, Henry, Hoffmann, Michael +1
#05C62 (Secondary) #68R10 (Primary) 05C10 #Combinatorics (math.CO) #Computational Geometry (cs.CG) #Discrete Mathematics (cs.DM) #F.2.2 #FOS: Computer and information sciences #FOS: Mathematics #G.2.1
paper · doi:10.48550/arxiv.2408.14299
We show that every planar graph has a monotone topological 2-page book embedding where at most (4n-10)/5 (of potentially 3n-6) edges cross the spine, and every edge crosses the spine at most once; such an edge is called a biarc. We can also guarantee that all edges that cross the spine cross it in the same direction (e.g., from bottom to top). For planar 3-trees we can further improve the bound to (3n-9)/4, and for so-called Kleetopes we obtain a bound of at most (n-8)/3 edges that cross the spine. The bound for Kleetopes is tight, even if the drawing is not required to be monotone. A Kleetope is a plane triangulation that is derived from another plane triangulation T by inserting a new vertex vf into each face f of T and then connecting vf to the three vertices of f.