Geometric dominating sets - a minimum version of the No-Three-In-Line Problem

Abstract

We consider a minimizing variant of the well-known No-Three-In-Line Problem, the Geometric Dominating Set Problem: What is the smallest number of points in an $n\times n$ grid such that every grid point lies on a common line with two of the points in the set? We show a lower bound of $\Omega \left(n^{2/3}\right)$ points and provide a constructive upper bound of size $2\lceil n/2\rceil$. If the points of the dominating sets are required to be in general position we provide optimal solutions for grids of size up to $12\times 12$. For arbitrary n the currently best upper bound for points in general position remains the obvious 2n. Finally, we discuss the problem on the discrete torus where we prove an upper bound of $O\left({(n\log n)}^{1/2}\right)$. For n even or a multiple of 3, we can even show a constant upper bound of 4. We also mention a number of open questions and some further variations of the problem.