Unit-length Rectangular Drawings of Graphs

TL;DR

The paper investigates the existence and recognition of unit-length rectangular (and inner-rectangular) drawings of planar graphs, highlighting a sharp complexity boundary driven by outer-face constraints. It proves NP-hardness for both unit-length inner-rectangular and rectangular drawing recognition when the outer face need not be a rectangle (even with fixed embeddings) and provides linear-time, cubic-time, and linear-time-with-constraints algorithms under outer-face rectangle conditions and degree restrictions, respectively. Using SPQR-tree analysis, it derives structural conditions that enable efficient (cubic or linear) solutions and, as a by-product, gives the first polynomial-time algorithm for recognizing rectangular drawings in graphs with maximum degree $4$. Overall, the work delineates a clear dichotomy between cases with and without outer-face rectangle requirements and advances both theory and practical algorithms for unit-length orthogonal rectangle representations.

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.

Figures (2)

• Figure 1: Unit-length embedding-preserving rectangular drawings of a plane graph.
• Figure 2: \ref{['fig:grid-drawing-a']} A planar rectilinear grid drawing of a graph. \ref{['fig:grid-drawing-b']} A unit-length rectangular grid drawing of the same graph.