(t,r) broadcast domination in the infinite grid

TL;DR

This paper investigates the minimal density of towers needed for broadcast domination in infinite grids, proves a conjecture for certain parameters, and calculates domination numbers for specific graph classes.

Contribution

It proves a conjecture on tower density for $(t,3)$ broadcast domination in $ ext{Z}^2$ for large $t$, and computes $(t,r)$ domination numbers for powers of paths and cycles.

Findings

Proved minimal tower density for $(t,3)$ broadcast in $ ext{Z}^2$ for $t>17$

Determined $(t,r)$ broadcast domination numbers for powers of paths and cycles

Explored generalizations of broadcast domination in infinite grids

Abstract

The $(t,r)$ broadcast domination number of a graph $G$, $\gamma_{t,r}(G)$, is a generalization of the domination number of a graph. $\gamma_{t,r}(G)$ is the minimal number of towers needed, placed on vertices of $G$, each transmitting a signal of strength $t$ which decays linearly, such that every vertex receives a total amount of at least $r$ signal. In this paper we prove a conjecture by Drews, Harris, and Randolph about the minimal density of towers in $\mathbb{Z}^2$ that provide a $(t,3)$ domination broadcast for $t>17$ and explore generalizations. Additionally, we determine the $(t,r)$ broadcast domination number of powers of paths, $P_n^{(k)}$ and powers of cycles, $C_n^{(k)}$.