Monotone Arc Diagrams with few Biarcs

We show that every planar graph has a monotone arc diagram, also known as 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 further improve the bound to (3n−9)/4, and for so-called Kleetopes we obtain a bound of at most (n − 8)/3 biarcs. 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 v_f into each face f of T and then connecting v_f to the three vertices of f.