Rectangle-visibility representations of bipartite graphs

Abstract

The paper considers representations of bipartite graphs as rectangle-visibility graphs, i.e., graphs whose vertices are rectangles in the plane, with adjacency determined by horizontal and vertical visibility. It is shown that, for p ≤ q, Kp,q has a representation with no rectangles having collinear sides if and only if p < 3 or p = 3 and q ≤ 4. More generally, it is shown that Kp,q is a rectangle-visibility graph if and only if p ≤ 4. Finally, it is shown that every bipartite rectangle-visibility graph on n ≥ 4 vertices has at most 4n – 12 edges.