On Rectangle Visibility Graphs

In this paper we consider the problem of drawing a graph in the plane so that the vertices of the graph are drawn as rectangles and the edges are horizontal or vertical line segments. We are in particular interested in drawings where each of the line segments can be thickened to have positive width, and none of these thickened segments (which we call bands of visibility) intersects the interior of any of the rectangles , although the bands may themselves cross or intersect. We call a graph a rectangle-visibility graph (or RVG for short) if it has such a drawing; we call the drawing itself the layout of the graph. We study three variations on this idea. In the rst, we require that the graph be drawn so that every pair of rectangles with a possible band of visibility between them represents a pair of vertices joined by an edge in the graph, and furthermore that no two rectangles have sides contained in the same (horizontal or vertical) line; we call such graphs noncollinear RVGs. In the second, we still require that every possible visibility band represents an edge, but we allow two rectangles to have collinear edges; we call such graphs RVGs or collinear RVGs for emphasis. In the third, we do not require that every possible visibility band represents an edge; we call such graphs weak RVGs. Collinearities in drawings of weak RVGs can be eliminated by perturbation; thus, the collinear/noncollinear distinction is actually unnecessary. Every noncollinear RVG is a collinear RVG; it is also true that every collinear RVG is a weak RVG. Furthermore there are weak RVGs that are not collinear RVGs, and collinear RVGs that are not non-collinear. These results are formally shown in Dean and Hutchinson 1]. We will only brieey review some related work about RVGs. For a more thorough treatment, as well as motivational details, the reader is referred to the full paper. Wismath 6] seems to be the rst to have studied RVGs, and showed that every planar graph is a collinear RVG. Hutchinson, Shermer, and Vince 3] established that no RVG has more than 6n ? 20 edges (where n is the number of vertices). Dean and Hutchinson 1] determined which complete bipar-tite graphs are collinear RVGs (K p;q for p 4) and noncollinear RVGs (K p;q for p 2 or p; q = (3; 3) or p; q = …