Abstract
This paper studies problems related to visibility among points in the plane. A point xblocks two points v and w if x is in the interior of the line segment \(\overline{vw}\). A set of points P is k-blocked if each point in P is assigned one of k colors, such that distinct points v, w ∈ P are assigned the same color if and only if some other point in P blocks v and w. The focus of this paper is the conjecture that each k-blocked set has bounded size (as a function of k). Results in the literature imply that every 2-blocked set has at most 3 points, and every 3-blocked set has at most 6 points. We prove that every 4-blocked set has at most 12 points, and that this bound is tight. In fact, we characterize all sets \(\{{n}_{1},{n}_{2},{n}_{3},{n}_{4}\}\) such that some 4-blocked set has exactly n i points in the ith color class. Among other results, for infinitely many values of k, we construct k-blocked sets with k 1. 79… points.
This is the full version of an extended abstract presented at the 26th European Workshop on Computational Geometry (EuroCG’10).
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Notes
- 1.
If G is the visibility graph of some point set \(P \subseteq {\mathbb{R}}^{d}\), then G is the visibility graph of some projection of P to \({\mathbb{R}}^{2}\) (since a random projection of P to \({\mathbb{R}}^{2}\) is occlusion-free with probability 1).
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Acknowledgements
This research was initiated at The 24th Bellairs Winter Workshop on Computational Geometry, held in February 2009 at the Bellairs Research Institute of McGill University in Barbados. The authors are grateful to Godfried Toussaint and Erik Demaine for organizing the workshop, and to the other participants for providing a stimulating working environment. Thanks to the reviewer for many helpful suggestions. David R. Wood was supported by a QEII Research Fellowship from the Australian Research Council.
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Aloupis, G., Ballinger, B., Collette, S., Langerman, S., Pór, A., Wood, D.R. (2013). Blocking Colored Point Sets. In: Pach, J. (eds) Thirty Essays on Geometric Graph Theory. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0110-0_4
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