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：平面图的矩形绘图 $G$是一张平面图,其中将脊椎绘制成网格点,将边缘绘制成水平和垂直直线段,将面部画成矩形。有时,外部面的后一限制会放松。在本文中,我们研究边缘有单位长度的矩形图。我们显示了决定单长矩形绘图存在问题的复杂二分法,取决于外部面是否也必须画成矩形。具体地说,我们证明,当外部面的绘图不需要成为矩形时,即使所要绘制的绘图必须尊重给定的平面嵌入,而它是在固定和变量嵌入设置中,如果需要将外部面画成矩形,则问题就是NP的完整。