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.

翻译：平面图$G$的矩形绘制是指$G$的一种平面绘制方式，其中顶点映射至网格点，边映射为水平或垂直的直线段，面被绘制为矩形。有时外层面的这一约束会被放宽。本文研究边长为单位长度的矩形绘制。我们根据外层面是否必须绘制为矩形，证明了判定单位长度矩形绘制存在性问题的复杂度二分性。具体而言，我们证明了对于双连通图，当不要求外层面绘制为矩形时，即使所求绘制必须遵循给定的平面嵌入，该问题也是NP完全的；而若要求外层面绘制为矩形，则在固定嵌入和可变嵌入两种设置下，该问题均存在多项式时间解法。