  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.
