# Inserting One Edge into a Simple Drawing is Hard

Alan Arroyo, Fabian Klute, Irene Parada*, Birgit Vogtenhuber, Raimund Seidel, Tilo Wiedera

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

## Abstract

A simple drawing D(G) of a graph G is one where each pair of edges share at most one point: either a common endpoint or a proper crossing. An edge e in the complement of G can be inserted into D(G) if there exists a simple drawing of $$G+e$$extending D(G). As a result of Leviâ€™s Enlargement Lemma, if a drawing is rectilinear (pseudolinear), that is, the edges can be extended into an arrangement of lines (pseudolines), then any edge in the complement of G can be inserted. In contrast, we show that it is NP-complete to decide whether one edge can be inserted into a simple drawing. This remains true even if we assume that the drawing is pseudocircular, that is, the edges can be extended to an arrangement of pseudocircles. On the positive side, we show that, given an arrangement of pseudocircles $$\mathcal {A}$$and a pseudosegmentÂ $$\sigma$$, it can be decided in polynomial time whether there exists a pseudocircle $$\Phi _\sigma$$extending $$\sigma$$for which $$\mathcal {A}\cup \{\Phi _\sigma \}$$is again an arrangement of pseudocircles.
Original language English Discrete & Computational Geometry https://doi.org/10.1007/s00454-022-00394-9 E-pub ahead of print - 2022

## Fields of Expertise

• Information, Communication & Computing

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