Area, Perimeter, Height, and Width of Rectangle Visibility Graphs
John S. Caughman, Charles L. Dunn, Joshua D. Laison, Nancy Ann, Neudauer, Colin L. Starr

TL;DR
This paper investigates the geometric parameters of rectangle visibility graphs, demonstrating their independence and analyzing how different graph structures influence minimal bounding box measures such as area, perimeter, height, and width.
Contribution
It introduces the first comprehensive analysis of how area, perimeter, height, and width parameters vary independently in rectangle visibility graphs, including constructions and bounds.
Findings
These four measures are mutually independent for RVGs.
Existence of graphs requiring different representations to minimize each parameter.
Complete graphs with 7 or 8 vertices require larger area than empty graphs.
Abstract
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, width, or height (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 and with but , and analogously for all other pairs of these parameters. We further show that there exists a graph with…
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Robotic Path Planning Algorithms
