Visibility Representations of Complete Graphs

Abstract

In this paper we study 3-dimensional visibility representations of complete graphs. The vertices are represented by equal regular polygons lying in planes parallel to the xy-plane. Two vertices are adjacent if and only if the two corresponding polygons see each otheri.e. it is possible to construct an abscissa perpendicular to the xy-plane connecting the two polygons and avoiding all the others. We give the bounds for the maximal size f (k) of a clique represented by regular k-gons: k+1 2 + 2 ≤ f (k) ≤ 2 2 k and we present a particular result for triangles: f (3) ≥ 14.

3D Visibility Representations of Complete Graphs

“…Let's remind the notion of polygon coordinates in a visibility representation (introduced in [2]). Firstly note that because of the openness of the polygons used in the representation we can expect that no two projections (into the plane z = 0) of sides of the polygons lie in the same line.…”

“…They also describe a representation of any graph on n vertices by (possibly different) polygons with at most 2n sides. Babilon et al [2] are interested in the maximum size of a clique represented by regular n-gons and present a lower bound n+1 2 + 2. Cobos et al [4] extend visibility representations even into the higher dimensions and characterize the dimensionality (e.g.…”

Vertical Visibility Among Parallel Polygons in Three Dimensions

“…Hence, lim k→∞ f (k) = ∞. Fekete et al [8] proved that f (4) = 7 thereby showing that f (k) is not monotone in k. Nevertheless, it is plausible that f (k + 2) ≥ f (k) for every k, and surprisingly enough this is stated as an open problem in [1]. Another interesting open problem from the same paper is to decide if the limit lim k→∞ f (k) k exists.…”

“…In the present note we improve the above upper bound on f (k) to O(k 4 ) 3 and we extend our investigation to families of homothetes of general convex polygons. The main tool to obtain the result is Dilworth Theorem [6], which was also used by Babilon et al to obtain the doubly exponential bound in [1]. Roughly speaking, our improvement is achieved by applying Dilworth Theorem only once whereas Babilon et al used its k successive applications.…”

“…Babilon et al [1] studied the following three-dimensional visibility representations of complete graphs. The vertices are represented by translates of a regular convex polygon lying in distinct planes parallel to the xy-plane and two translates are joined by an edge if they can see each other, which happens if it is possible to connect them by a line segment orthogonal to the xy-plane avoiding all the other translates.…”