Blocking Colored Point Sets

Greg Aloupis (), Brad Ballinger (), Sébastien Collette (), Stefan Langerman (), Attila Pór () and David R. Wood ()
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Greg Aloupis: Université Libre de Bruxelles, Chargé de Recherches du F.R.S.-FNRS, Département d’Informatique
Brad Ballinger: Humboldt State University, Department of Mathematics
Sébastien Collette: Université Libre de Bruxelles, Chargé de Recherches du F.R.S.-FNRS, Département d’Informatique
Stefan Langerman: Université Libre de Bruxelles, Maître de Recherches du F.R.S.-FNRS, Département d’Informatique
Attila Pór: Western Kentucky University, Department of Mathematics
David R. Wood: The University of Melbourne, Department of Mathematics and Statistics

A chapter in Thirty Essays on Geometric Graph Theory, 2013, pp 31-48 from Springer

Abstract: 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.

Keywords: Color Class; Visibility Graph; Monochromatic Pair; Motivational Background; Fundamental Conjecture (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/978-1-4614-0110-0_4

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