The Complexity of Drawing a Graph in a Polygonal Region

TL;DR

This paper proves that determining a straight-line planar graph drawing within a polygonal region is computationally complex, being complete for the existential theory of the reals, and explores the implications of coordinate irrationality.

Contribution

It establishes the problem's complexity class as complete for the existential theory of the reals and demonstrates the necessity of irrational coordinates in some cases.

Findings

The problem is complete for the existential theory of the reals.

Drawing may require irrational coordinates even with integer inputs.

Complexity remains open for simply connected regions.

Abstract

We prove that the following problem is complete for the existential theory of the reals: Given a planar graph and a polygonal region, with some vertices of the graph assigned to points on the boundary of the region, place the remaining vertices to create a planar straight-line drawing of the graph inside the region. This strengthens an NP-hardness result by Patrignani on extending partial planar graph drawings. Our result is one of the first showing that a problem of drawing planar graphs with straight-line edges is hard for the existential theory of the reals. The complexity of the problem is open in the case of a simply connected region. We also show that, even for integer input coordinates, it is possible that drawing a graph in a polygonal region requires some vertices to be placed at irrational coordinates. By contrast, the coordinates are known to be bounded in the special case of a convex region, or for drawing a path in any polygonal region.