Unit-length Rectangular Drawings of Graphs

TL;DR

This paper investigates the computational complexity of creating unit-length rectangular drawings of planar graphs, revealing NP-completeness in some cases and polynomial solvability in others based on outer face constraints.

Contribution

It establishes a complexity dichotomy for the problem, showing NP-completeness when the outer face isn't constrained and polynomial-time solutions when it is.

Findings

NP-complete for biconnected graphs without outer face rectangle constraint

Polynomial-time algorithms when the outer face is a rectangle

Complexity depends on outer face constraints

Abstract

A rectangular drawing of a planar graph $G$ is a planar drawing of $G$ in which vertices are mapped to grid points, edges are mapped to horizontal and vertical straight-line segments, and faces are drawn as rectangles. Sometimes this latter constraint is relaxed for the outer face. In this paper, we study rectangular drawings in which the edges have unit length. We show a complexity dichotomy for the problem of deciding the existence of a unit-length rectangular drawing, depending on whether the outer face must also be drawn as a rectangle or not. Specifically, we prove that the problem is NP-complete for biconnected graphs when the drawing of the outer face is not required to be a rectangle, even if the sought drawing must respect a given planar embedding, whereas it is polynomial-time solvable, both in the fixed and the variable embedding settings, if the outer face is required to be drawn as a rectangle.