Area, perimeter, height, and width of rectangle visibility graphs

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.