New Results On A Visibility Representation Of Graphs In 3d

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); rye 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 (K-7 can be represented but K-n for n > 7 cannot);- a representation of K-n by unit discs.