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On $(t,r)$ broadcast domination of certain grid graphs

### Natasha Crepeau, Pamela E. Harris, Sean Hays, Marissa Loving, Joseph Rennie, Gordon Rojas Kirby and Alexandro Vasquez

Vol. 16 (2023), No. 5, 883–903
##### Abstract

Let $G=\left(V\left(G\right),E\left(G\right)\right)$ be a connected graph with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$. We say a subset $D$ of $V\left(G\right)$ dominates $G$ if every vertex in $V\setminus D$ is adjacent to a vertex in $D$. A generalization of this concept is $\left(t,r\right)$ broadcast domination. We designate certain vertices to be towers of signal strength $t$, which send out signal to neighboring vertices with signal strength decaying linearly as the signal traverses the edges of the graph. We let $\mathbb{𝕋}$ be the set of all towers, and we define the signal received by a vertex $v\in V\left(G\right)$ from all towers $w\in \mathbb{𝕋}$ to be $f\left(v\right)={\sum }_{w\in \mathbb{𝕋}}\mathrm{max}\left(0,t-d\left(v,w\right)\right)$. Blessing, Insko, Johnson and Mauretour defined a $\left(t,r\right)$ broadcast dominating set, or a $\left(t,r\right)$ broadcast, on $G$ as a set $\mathbb{𝕋}\subseteq V\left(G\right)$ such that $f\left(v\right)\ge r$ for all $v\in V\left(G\right)$. The minimum cardinality of a $\left(t,r\right)$ broadcast on $G$ is called the $\left(t,r\right)$ broadcast domination number of $G$. We present our research on the $\left(t,r\right)$ broadcast domination number for certain graphs including paths, grid graphs, the slant lattice, and the king’s lattice.

##### Keywords
graph domination, grid graphs
##### Mathematical Subject Classification
Primary: 05C12, 05C30, 05C69
##### Milestones
Received: 23 August 2022
Accepted: 1 December 2022
Published: 9 December 2023

Communicated by Joel Foisy
##### Authors
 Natasha Crepeau Department of Mathematics University of Washington Seattle, WA United States Pamela E. Harris Department of Mathematical Sciences University of Wisconsin-Milwaukee Milwaukee, WI United States Sean Hays Department of Mathematics The University of Alabama Tuscaloosa, AL United States Marissa Loving Department of Mathematics University of Wisconsin-Madison Madison, WI United States Joseph Rennie Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL United States Gordon Rojas Kirby School of Mathematics and Statistical Sciences Arizona State University Tempe, AZ United States Alexandro Vasquez Department of Mathematics Manhattan College Riverdale, NY 10471 United States
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