Bounds On $(t,r)$ Broadcast Domination of $n$-Dimensional Grids

TL;DR

This paper investigates the density bounds of $(t,r)$ broadcast domination in infinite grids, introduces methods for calculating these bounds, and presents counterexamples to a generalized Vizing's Conjecture for $r \,\ge\, 2$.

Contribution

It provides new bounds and computational methods for $(t,r)$ broadcast domination in grids and challenges a generalized Vizing's Conjecture with counterexamples.

Findings

Established upper and lower bounds on domination density.

Developed methods to compute these bounds.

Presented counterexamples to the generalized Vizing's Conjecture for $r \ge 2$.

Abstract

In this paper, we study a variant of graph domination known as $(t, r)$ broadcast domination, first defined in Blessing, Insko, Johnson, and Mauretour in 2015. In this variant, each broadcast provides $t-d$ reception to each vertex a distance $d < t$ from the broadcast. If $d \ge t$ then no reception is provided. A vertex is considered dominated if it receives $r$ total reception from all broadcasts. Our main results provide some upper and lower bounds on the density of a $(t, r)$ dominating pattern of an infinite grid, as well as methods of computing them. Also, when $r \ge 2$ we describe a family of counterexamples to a generalization of Vizing's Conjecture to $(t,r)$ broadcast domination.