Monotone Arc Diagrams with few Biarcs
<|reference_start|>Monotone Arc Diagrams with few Biarcs: 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 v_f into each face f of T and then connecting v_f to the three vertices of f.<|reference_end|>
@article{chaplick2024monotone,
    title={Monotone Arc Diagrams with few Biarcs},
    author={Steven Chaplick and Henry F"orster and Michael Hoffmann and Michael
  Kaufmann},
    journal={arXiv preprint arXiv:2408.14299},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.14299},
    primaryClass={cs.DM cs.CG math.CO}
}