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.
Research supported by grant GAUK 159/99.
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H. Alt, M. Godau, S. Whitesides: Universal 3-Dimensional Visibility Representations for Graphs Proc. Graph Drawing’ 95, Passau, 1995. Lecture Notes in Computer Science LNCS 1027, Springer-Verlag, 1996, pp. 8–19
S. P. Fekete, M. E. Houle, S. Whitesides: New Results on a Visibility Representation of Graphs in 3D Proc. Graph Drawing’ 95, Passau, 1995. Lecture Notes in Computer Science LNCS 1027, Springer-Verlag, 1996, pp. 234–241
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Babilon, R., Nyklová, H., Pangrác, O., Vondrák, J. (1999). Visibility Representations of Complete Graphs. In: Kratochvíyl, J. (eds) Graph Drawing. GD 1999. Lecture Notes in Computer Science, vol 1731. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46648-7_34
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DOI: https://doi.org/10.1007/3-540-46648-7_34
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