The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings

Many well-known graph drawing techniques, including force directed drawings, spectral graph layouts, multidimensional scaling, and circle packings, have algebraic formulations. However, practical methods for producing such drawings ubiquitously use iterative numerical approximations rather than constructing and then solving algebraic expressions representing their exact solutions. To explain this phenomenon, we use Galois theory to show that many variants of these problems have solutions that cannot be expressed by nested radicals or nested roots of low-degree polynomials. Hence, such solutions cannot be computed exactly even in extended computational models that include such operations.Comment: Graph Drawing 201

Circle-Representations of Simple 4-Regular Planar Graphs

Lovász conjectured that every connected 4-regular planar graph G admits a realization as a system of circles, i.e., it can be drawn on the plane utilizing a set of circles, such that the vertices of G correspond to the intersection and touching points of the circles and the edges of G are the arc segments among pairs of intersection and touching points of the circles. In this paper, (a) we affirmatively answer Lovász's conjecture, if G is 3-connected, and, (b) we demonstrate an infinite class of connected 4-regular planar graphs which are not 3-connected and do not admit a realization as a system of circles

On a conjecture of Lovász on circle-representations of simple 4-regular planar graphs

Lovász conjectured that every connected 4-regular planar graph $G$ admits a realization as a system of circles, i.e., it can be drawn on the plane utilizing a set of circles, such that the vertices of $G$ correspond to the intersection and touching points of the circles and the edges of $G$ are the arc segments among pairs of intersection and touching points of the circles. In this paper, we settle this conjecture. In particular, (a) we first provide tight upper and lower bounds on the number of circles needed in a realization of any simple 4-regular planar graph, (b) we affirmatively answer Lovász's conjecture, if $G$ is 3-connected, and, (c) we demonstrate an infinite class of simple connected 4-regular planar graphs which are not 3-connected (i.e., either simply connected or biconnected) and do not admit realizations as a system of circles