Finite Square Lattice Vertex Cover by a Baseline Set Defined With a Minimum Sublattice

TL;DR

This paper studies the minimum vertex subset in a finite square lattice that covers all points via lines defined by the subset's edges, with applications in optimizing signal transmission in square arrays.

Contribution

It introduces a novel approach to identify minimal vertex sets that cover all lattice points through line-based coverage, extending the classical vertex cover problem.

Findings

Identifies minimum vertex subsets for complete lattice coverage.

Provides bounds and algorithms for the minimal cover sets.

Demonstrates applications in signal distribution in square arrays.

Abstract

Each straight infinite line defined by two vertices of a finite square point lattice contains (covers) these two points and a - possibly empty - subset of points that happen to be collinear to these. This work documents vertex subsets of minimum order such that the sum of the infinite straight lines associated with the edges of their complete subgraph covers the entire set of vertices (nodes). This is an abstraction to the problem of sending a light signal to all stations (receivers) in a square array with a minimum number of stations also equipped with transmitters to redirect the light to other transmitters.