Lecture Notes in Computer Science Volume 1731, 1999, pp 333-340

Visibility Representations of Complete Graphs

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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 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: \( \left\lfloor {\tfrac{{k + 1}} {2}} \right\rfloor + 2 \leqslant f(k) \leqslant 2^{2^k } \) and we present a particular result for triangles: f(3) ≥ 14.