Blocking Visibility for Points in General Position

Abstract

For a finite set P in the plane, let b(P ) be the smallest possible size of a set Q, Q ∩ P = ∅, such that every segment with both endpoints in P contains at least one point of Q. We raise the problem of estimating b(n), the minimum of b(P ) over all n-point sets P with no three points collinear. We review results providing bounds on b(n) and mention some additional observations.Let P be an n-point set in the plane (or, more generally, in R d ). We define a visibilityblocking set for P as a set Q that is disjoint from P and such that every segment with endpoints in P contains at least one point of Q.If the points of P are all collinear, then there is a visibility blocking set with n − 1 points. The question raised in this note is, what is the smallest possible size of a visibility-blocking set for P having no three points collinear? That is, we let b(P ) := min{|Q| : Q a visibility-blocking set for P} b(n) := min{b(P ) : P ⊂ R 2 with no three points collinear, |P | = n}, and we would like to estimate the asymptotics of b(n) for large n.

Blocking Colored Point Sets

“…If S is the largest colour class then P − S blocks S. By Lemma 4, S is in general position. By the results of Matoušek [7] and Dumitrescu et al [2] mentioned in Section 3, n − m ≥ 2m − 3 and n − m ≥ (…”

“…Matoušek [7] proved that b(n) ≥ 2n − 3. Dumitrescu et al [2] improved this bound to b(n) ≥ ( n → ∞ as n → ∞.…”

“…Thus b(n) ≤ m(n) ≤ cn log 3 = cn 1.585... . This upper bound was improved by Pach [8] (and later by Matoušek [7]) to…”

Every Large Point Set contains Many Collinear Points or an Empty Pentagon

“…• ℓ collinear points, or • k pairwise visible points (that is, the visibility graph contains a k-clique). [1,23,27]. It is trivially true for ℓ ≤ 3 and all k. Kára et al [23] proved it for k ≤ 4 and all ℓ. Addario-Berry et al [1] proved it in the case that k = 5 and ℓ = 4.…”

Unsolved problems in visibility graphs of points, segments, and polygons

“…b(n) ≥ n−1. A better lower bound can be derived using a triangulation of P [78]. Observe that every edge of a triangulation must contain one blocker.…”