# Extending Simple Drawings

** Published:**

Recommended citation: Arroyo, A., Bensmail, J. and Richter, R.B., 2018. Extending Drawings of Graphs to Arrangements of Pseudolines. arXiv preprint arXiv:1804.09317. __https://arxiv.org/pdf/1908.08129.pdf__

Simple drawings of graphs are those in which each pair of edges share at most one point, either a common endpoint or a proper crossing. In this paper we study the problem of extending a simple drawing of a graph by inserting a set of edges from the complement of into such that the result is a simple drawing. In the context of rectilinear drawings, the problem is trivial. For pseudolinear drawings, the existence of such an extension follows from Levi’s enlargement lemma. In contrast, we prove that deciding if a given set of edges can be inserted into a simple drawing is NP-complete. Moreover, we show that the maximization version of the problem is APX-hard. We also present a polynomial-time algorithm for deciding whether one edge uv can be inserted into when is a dominating set for the graph .