On (t,r) Broadcast Domination Numbers of Grids

TL;DR

This paper introduces and determines exact values for a new family of domination parameters called (t,r) broadcast domination numbers on small grids, providing bounds and conjectures for larger grids.

Contribution

It defines the (t,r) broadcast domination numbers, generalizing existing domination concepts, and computes exact values for small grids along with bounds and conjectures for larger ones.

Findings

Exact (t,r) broadcast domination numbers for small grids

Upper bounds for large grids' (t,r) broadcast domination numbers

Conjecture that bounds are tight for large grids

Abstract

The domination number of a graph $G = (V,E)$ is the minimum cardinality of any subset $S \subset V$ such that every vertex in $V$ is in $S$ or adjacent to an element of $S$. Finding the domination numbers of $m$ by $n$ grids was an open problem for nearly 30 years and was finally solved in 2011 by Goncalves, Pinlou, Rao, and Thomass\'e. Many variants of domination number on graphs have been defined and studied, but exact values have not yet been obtained for grids. We will define a family of domination theories parameterized by pairs of positive integers $(t,r)$ where $1 \leq r \leq t$ which generalize domination and distance domination theories for graphs. We call these domination numbers the $(t,r)$ broadcast domination numbers. We give the exact values of $(t,r)$ broadcast domination numbers for small grids, and we identify upper bounds for the $(t,r)$ broadcast domination numbers for large grids and conjecture that these bounds are tight for sufficiently large grids.