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 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.