On the Complexity of Lombardi Graph Drawing

TL;DR

This paper proves that determining whether a general graph can be drawn as a Lombardi drawing with a fixed edge ordering is computationally very hard, establishing NP-hardness and $orall ext{-} ext{R}$-completeness.

Contribution

It is the first work to analyze the computational complexity of constructing Lombardi drawings for general graphs, showing the problem is NP-hard and $orall ext{-} ext{R}$-complete.

Findings

Determining Lombardi drawability with fixed edge orderings is $orall ext{-} ext{R}$-complete.

The problem is NP-hard for general graphs.

This is the first complexity result for Lombardi drawing existence in general graphs.

Abstract

In a Lombardi drawing of a graph the vertices are drawn as points and the edges are drawn as circular arcs connecting their respective endpoints. Additionally, all vertices have perfect angular resolution, i.e., all angles incident to a vertex $v$ have size $2\pi/\mathrm{deg}(v)$. We prove that it is $\exists\mathbb{R}$-complete to determine whether a given graph admits a Lombardi drawing respecting a fixed cyclic ordering of the incident edges around each vertex. In particular, this implies NP-hardness. While most previous work studied the (non-)existence of Lombardi drawings for different graph classes, our result is the first on the computational complexity of finding Lombardi drawings of general graphs.