Inserting an Edge into a Geometric Embedding

TL;DR

This paper studies the problem of inserting an edge into a geometric embedding of a graph to minimize crossings, providing algorithms for special cases and complexity results for the general problem.

Contribution

It introduces polynomial-time algorithms for specific cases and proves the fixed-parameter tractability of the general problem, along with an approximation method.

Findings

Polynomial-time algorithms for special cases

Fixed-parameter tractability of the general problem

Approximation of crossings within a factor based on maximum degree

Abstract

The algorithm of Gutwenger et al. to insert an edge $e$ in linear time into a planar graph $G$ with a minimal number of crossings on $e$, is a helpful tool for designing heuristics that minimize edge crossings in drawings of general graphs. Unfortunately, some graphs do not have a geometric embedding $\Gamma$ such that $\Gamma+e$ has the same number of crossings as the embedding $G+e$. This motivates the study of the computational complexity of the following problem: Given a combinatorially embedded graph $G$, compute a geometric embedding $\Gamma$ that has the same combinatorial embedding as $G$ and that minimizes the crossings of $\Gamma+e$. We give polynomial-time algorithms for special cases and prove that the general problem is fixed-parameter tractable in the number of crossings. Moreover, we show how to approximate the number of crossings by a factor $(\Delta-2)$, where $\Delta$ is the maximum vertex degree of $G$.