On the $2$-Bend Slope Number of $1$-Planar Graphs

Michael A. Bekos; Eleni Katsanou; Philipp Kindermann; Aikaterini Maria Ntasiou; Maria Eleni Pavlidi; Soeren Terziadis

On the $2$-Bend Slope Number of $1$-Planar Graphs

Computational Geometry 2026-07-28 v1

Authors: Michael A. Bekos , Eleni Katsanou , Philipp Kindermann , Aikaterini Maria Ntasiou , Maria Eleni Pavlidi , Soeren Terziadis

View on arXiv ↗ PDF ↗

Abstract

While drawing planar graphs with few slopes and few bends is a well-studied problem, corresponding extensions to beyond-planar graphs still remain mostly unexplored. Motivated by this observation, in this work, we provide bounds on the slope number of biconnected $1$ -planar graphs when two bends are allowed along each edge. Our contribution is an incremental drawing algorithm that produces $2$ -bend $1$ -planar drawings of biconnected $1$ -plane graphs with maximum degree $\Delta$ using any prescribed set of $\Delta$ pairwise distinct slopes.

Keywords

slope number 1-planar graphs graph drawing bends incremental algorithm biconnected graphs

Cite

@article{arxiv.2607.25553,
  title  = {On the $2$-Bend Slope Number of $1$-Planar Graphs},
  author = {Michael A. Bekos and Eleni Katsanou and Philipp Kindermann and Aikaterini Maria Ntasiou and Maria Eleni Pavlidi and Soeren Terziadis},
  journal= {arXiv preprint arXiv:2607.25553},
  year   = {2026}
}

Comments

A preliminary version of this work was published at GD 2026