# Geometric Dominating Sets

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× 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 Ω(n^2/3) points and provide a constructive upper bound of size 2 ⌈ n/2 ⌉. 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 × 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((n log n)^1/2). 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.

• 25 publications
• 67 publications
• 3 publications
05/13/2022

### Improved Upper Bound on Independent Domination Number for Hypercubes

We revisit the problem of determining the independent domination number ...
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### General Position Problem of Butterfly Networks

A general position set S is a set S of vertices in G(V,E) such that no t...
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### GridTuner: Reinvestigate Grid Size Selection for Spatiotemporal Prediction Models [Technical Report]

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### On 4-general sets in finite projective spaces

A 4-general set in PG(n,q) is a set of points of PG(n,q) spanning the wh...
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### Shifted varieties and discrete neighborhoods around varieties

For an affine variety X defined over a finite prime field F_p and some i...
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### No Selection Lemma for Empty Triangles

Let S be a set of n points in general position in the plane. The Second ...
06/09/2022

### Distinct Angles in General Position

The Erdős distinct distance problem is a ubiquitous problem in discrete ...