Visibility Representations of Complete Graphs
نویسندگان
چکیده
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 other i.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) ≤ 22k and we present a particular result for triangles: f(3) ≥ 14.
منابع مشابه
Representations of Graphs on a Cylinder
A complete characterization of the class of graphs that admit a cylindric visibility representation is presented, where vertices are represented by intervals parallel to the axis ofthe cylinder and the edges correspond to pairs of visible intervals. Moreover, linear time algorithms are given for testing the existence of and constructing such a representation. Important applications of cylindric...
متن کاملOn a Visibility Representation of Graphs
We give a visibility representation of graphs which extends some very well-known representations considered extensively in the literature. Concretely, the vertices are represented by a collection of parallel hyper-rectangles in R and the visibility is orthogonal to those hyperrectangles. With this generalization, we can prove that each graph admits a visibility representation. But, it arises th...
متن کاملBar k-Visibility Graphs: Bounds on the Number of Edges, Chromatic Number, and Thickness
Let S be a set of horizontal line segments, or bars, in the plane. We say that G is a bar visibility graph, and S its bar visibility representation, if there exists a one-to-one correspondence between vertices of G and bars in S, such that there is an edge between two vertices in G if and only if there exists an unobstructed vertical line of sight between their corresponding bars. If bars are a...
متن کاملBar k-Visibility Graphs
Let S be a set of horizontal line segments, or bars, in the plane. We say that G is a bar visibility graph, and S its bar visibility representation, if there exists a one-to-one correspondence between vertices of G and bars in S, such that there is an edge between two vertices in G if and only if there exists an unobstructed vertical line of sight between their corresponding bars. If bars are a...
متن کاملRectangle-Visibility Representations of Bipartite Graphs
The paper considers representations of bipartite graphs as rectangle-visibility graphs, Le., graphs whose vertices are rectangles in the plane, with adjacency detenmned by horizontal and vertical visibility.1t is shown that, for p ~ q, Kp,q has a representation with no rectangles having collinear sides if and only if p ~ 2 or p = 3 and q ~ 4. More generally, it is shown that Kp;q is a rectangle...
متن کامل