Extending Simple Drawings

TL;DR

This paper investigates the complexity of extending simple graph drawings by adding edges, proving NP-completeness and APX-hardness results, and providing a polynomial-time solution for certain cases.

Contribution

It establishes the NP-completeness and APX-hardness of extending simple graph drawings, and offers a polynomial-time algorithm for inserting edges when the endpoints form a dominating set.

Findings

Deciding edge insertion is NP-complete.

Maximization version is APX-hard.

Polynomial-time algorithm for inserting edges with endpoints as a dominating set.

Abstract

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 $D(G)$ of a graph $G$ by inserting a set of edges from the complement of $G$ into $D(G)$ 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 $D(G)$ when $\{u,v\}$ is a dominating set for the graph $G$.