New results on a visibility representation of graphs in 3D

Abstract

This paper considers a 3-dimensional visibility representation of cliques K n. In this representation, the objects representing the vertices are 2-dimensional and lie parallel to the x, y-plane, and two vertices of the graph are adjacent if and only if their corresponding objects see each other by a line of sight parallel to the z-axis that intersects the interiors of the objects. In particular, we represent vertices by unit discs and by discs of arbitrary radii (possibly different for different vertices); we also represent vertices by axis-aligned unit squares, by axis-aligned squares of arbitrary size (possibly different for different vertices), and by axis-aligned rectangles. We present: a significant improvement (from 102 to 55) of the best known upper bound for the size of cliques representable by rectangles or squares of arbitrary size; a sharp bound for the representation of cliques by unit squares (K7 can be represented but Kn for n>7 cannot); a representation of Kn by unit discs.