A rectangle visibility graph (RVG) is represented by assigning to each vertex a rectangle in the plane with horizontal and vertical sides in such a way that edges in the graph correspond to unobstructed horizontal and vertical lines of sight between their corresponding rectangles. To discretize, we consider only rectangles whose corners have integer coordinates. For any given RVG, we seek a representation with smallest bounding box as measured by its area, perimeter, height, or width (height is assumed not to exceed width). We derive a number of results regarding these parameters. Using these results, we show that these four measures are distinct, in the sense that there exist graphs \(G_1\) and \(G_2\) with \({{\,\textrm{area}\,}}(G_1) < {{\,\textrm{area}\,}}(G_2)\) but \({{\,\textrm{perim}\,}}(G_2) < {{\,\textrm{perim}\,}}(G_1)\), and analogously for all other pairs of these parameters. We further show that there exists a graph \(G_3\) with representations \(S_1\) and \(S_2\) such that \({{\,\textrm{area}\,}}(G_3)={{\,\textrm{area}\,}}(S_1)<{{\,\textrm{area}\,}}(S_2)\) but \({{\,\textrm{perim}\,}}(G_3)={{\,\textrm{perim}\,}}(S_2)<{{\,\textrm{perim}\,}}(S_1)\). In other words, \(G_3\) requires distinct representations to minimize area and perimeter. Similarly, such graphs exist to demonstrate the independence of all other pairs of these parameters. Among graphs with \(n \le 6\) vertices, the empty graph \(E_n\) requires largest area. But for graphs with \(n=7\) and \(n=8\) vertices, we show that the complete graphs \(K_7\) and \(K_8\) require larger area than \(E_7\) and \(E_8\), respectively. Using this, we show that for all \(n \ge 8\), the empty graph \(E_n\) does not have largest area, perimeter, height, or width among all RVGs on n vertices.

矩形可见性图（RVG）是通过在具有水平和垂直边的平面中为每个顶点分配一个矩形来表示的，使得图中的边缘对应于其相应矩形之间无障碍的水平和垂直视线。为了离散化，我们只考虑角点具有整数坐标的矩形。对于任何给定的 RVG，我们寻求具有最小边界框的表示，通过其面积、周长、高度或宽度来测量（假设高度不超过宽度）。我们得出了有关这些参数的许多结果。使用这些结果，我们表明这四个度量是不同的，因为存在图\(G_1\)和\(G_2\)以及\({{\,\textrm{area}\,}}(G_1) < {{\,\textrm{区域}\,}}(G_2)\)但\({{\,\textrm{perim}\,}}(G_2) < {{\,\textrm{perim}\,}}(G_1)\) ，对于这些参数的所有其他对也类似。我们进一步证明，存在一个图\(G_3\)，其表示为\(S_1\)和\(S_2\)，使得\({{\,\textrm{area}\,}}(G_3)={{\ ,\textrm{区域}\,}}(S_1)<{{\,\textrm{区域}\,}}(S_2)\) 但 \({{\,\textrm{perim}\,}} ( G_3 )={{\,\textrm{perim}\,}}(S_2)<{{\,\textrm{perim}\,}}(S_1)\)。换句话说，\(G_3\)需要不同的表示来最小化面积和周长。同样，此类图表的存在是为了证明这些参数的所有其他对的独立性。在具有\(n \le 6\)的图形中顶点，空图\(E_n\)需要最大面积。但是对于具有\(n=7\)和\(n=8\)顶点的图，我们表明完整的图\(K_7\)和\(K_8\)需要比\(E_7\)和\(分别为E_8\)。使用此方法，我们表明对于所有\(n \ge 8\)，空图\(E_n\)在n 个顶点上的所有 RVG 中不具有最大面积、周长、高度或宽度。