On a visibility representation for graphs in three dimensions

This paper proposes a 3-dimensional visibility representation of graphs G = (V,E) in which vertices are mapped to rectangles floating in R 3 parallel to the x, y-plane, with edges represented by vertical lines of sight. We apply an extension of the Erdős-Szekeres Theorem in a geometric setting to obtain an upper bound of n = 56 for the largest representable complete graph Kn. On the other hand, we show by construction that n ≥ 22. These are the best existing bounds. We also note that planar graphs and complete bipartite graphs Km,n are representable, but that the family of representable graphs is not closed under graph minors

On a visibility representation for graphs in 3D

This paper proposes a 3-dimensional visibility representation of graphs G = (V,E) in which vertices are mapped to rectangles floating in R 3 parallel to the x,y-plane, with edges represented by vertical lines of sight. We apply an extension of the Erdös-Szekeres Theorem in a geometric setting to obtain an upper bound of n = 56 for the largest representable complete graph Kn. On the other hand, we show by construction that n>=22. These are the best existing bounds. We also note that planar graphs and complete bipartite graphs Km,n are representable, but that the family of representable graphs is not closed under graph minors