3D Visibility Representations of Complete Graphs

Abstract

Abstract. This paper continues the study of 3D visibility representations of complete graphs where vertices are represented by equal convex polygons lying in planes parallel to the xy-plane. Edges correspond to the z-parallel visibility among these polygons. We give several bounds on the size of the largest complete graph that has a 3D visibility representation with particular properties. Namely we improve the best known lower bound for representations by regular ngons from

3D Visibility Representations by Regular Polygons

“…We show that the maximum size of a complete graph with a 3D visibility representation by regular n-gons is O(n 4 ). This result is a significant improvement of the previously known exponential bound 6n−3 3n−1 ¡ − 3 ≈ 2 6n from [1]. We don't attempt to minimize constants in this estimate because there still remains a big gap between the lower bound Ω(n) and our upper bound O(n 4 ).…”

“…Let {P 1 , P 2 , P 3 , P 4 } be a short-distance set of regular (2k + 1)-gons. If {P i } is a 3D visibility representation of a complete graph K 4 then it cannot happen that c 1 1 < c 2 1 < c 3 1 < c 4 1 and c 1 2 > c 2 2 > c 3 2 > c 4 2 (where (c i j ) n j=1 are coordinates of P i ).…”

“…Let {P 1 , P 2 , P 3 , P 4 } be a short-distance set of regular (2k + 1)gons. If {P i } is a 3D visibility representation of a complete graph K 4 then it cannot happen that c 1…”

“…The first upper bound 2 2 n was given by Babilon et al [2]. This doubly-exponential estimate was improved byŠtola [1] to an exponential 6n−3 3n−1 ¡ − 3 ≈ 2 6n . The main result of this paper is another significant improvement of this bound.…”

“…They also present a lower bound n+1 2 + 2 on the maximum size of a complete graph with a 3D visibility representation by equal regular n-gons. Stola [1] then moved this bound to n+1. The first upper bound 2 2 n was given by Babilon et al [2].…”