We introduce the $k$-Plane Insertion into Plane drawing ($k$-PIP) problem: given a plane drawing of a planar graph $G$ and a set of edges $F$, insert the edges in $F$ into the drawing such that the resulting drawing is $k$-plane. In this paper, we focus on the $1$-PIP scenario. We present a linear-time algorithm for the case that $G$ is a triangulation, while proving NP-completeness for the case that $G$ is biconnected and $F$ forms a path or a matching.

翻译：我们引入k-平面插入平面绘图（k-PIP）问题：给定平面图G的一个平面绘图和一组边F，将F中的边插入到该绘图中，使得结果绘图为k-平面的。本文重点研究1-PIP情形。对于G是三角剖分图的情况，我们提出了一种线性时间算法；同时，证明了当G是双连通图且F构成一条路径或一个匹配时，该问题是NP完全的。