On $k$-Plane Insertion into Plane Drawings
<|reference_start|>On $k$-Plane Insertion into Plane Drawings: We introduce the $k$-Plane Insertion into Plane drawing ($k$-PIP) problem: given a plane drawing of a planar graph $G$ and a set $F$ of edges, insert the edges in $F$ into the drawing such that the resulting drawing is $k$-plane. In this paper, we show that the problem is NP-complete for every $k\ge 1$, even when $G$ is biconnected and the set $F$ of edges forms a matching or a path. On the positive side, we present a linear-time algorithm for the case that $k=1$ and $G$ is a triangulation.<|reference_end|>
@article{katheder2024on,
    title={On $k$-Plane Insertion into Plane Drawings},
    author={Julia Katheder, Philipp Kindermann, Fabian Klute, Irene Parada, Ignaz
  Rutter},
    journal={arXiv preprint arXiv:2402.14552},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.14552},
    primaryClass={cs.CG}
}