Inserting one edge into a simple drawing is hard

TL;DR

Deciding whether a new edge can be added to a simple graph drawing is NP-complete, but extending arrangements of pseudocircles with a new pseudocircle can be decided efficiently.

Contribution

This paper proves NP-completeness for edge insertion in simple drawings and pseudocircular arrangements, and provides a polynomial-time algorithm for extending pseudocircle arrangements.

Findings

Edge insertion in simple drawings is NP-complete.

Extending pseudocircle arrangements with a new pseudocircle is polynomial-time solvable.

Levi's Enlargement Lemma applies to rectilinear and pseudolinear drawings, enabling edge insertion.

Abstract

A {\em 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 {\em inserted} into $D(G)$ if there exists a simple drawing of $G+e$ extending $D(G)$. As a result of Levi'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 pseudosegment $\sigma$, 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.