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Erschienen in:
2015 | OriginalPaper | Buchkapitel
Vertical Visibility Among Parallel Polygons in Three Dimensions
verfasst von : Radoslav Fulek, Rados Radoicic
Erschienen in: Graph Drawing and Network Visualization
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Abstract
Let \(\mathcal {C}=\{C_1,\ldots , C_n\}\) denote a collection of translates of a regular convex k-gon in the plane with the stacking order. The collection \(\mathcal {C}\) forms a visibility clique if for every \(i<j\) the intersection \(C_i\) and \(C_j\) is not covered by the elements that are stacked between them, i.e., \((C_i\cap C_j) \setminus \bigcup _{i<l<j}C_l\not =\emptyset \).
We show that if \(\mathcal {C}\) forms a visibility clique its size is bounded from above by \(O(k^4)\) thereby improving the upper bound of \(2^{2^{k}}\) from the aforementioned paper. We also obtain an upper bound of \(2^{2\left( {\begin{array}{c}k\\ 2\end{array}}\right) +2}\) on the size of a visibility clique for homothetes of a convex (not necessarily regular) k-gon.