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- W2772980790 abstract "Let $G=(V,E)$ be a graph and $t,r$ be positive integers. The signal that a vertex $v$ receives from a tower of signal strength $t$ located at vertex $T$ is defined as $sig(v,T)=max(t-dist(v,T),0)$, where $dist(v,T)$ denotes the distance between the vertices $v$ and $T$. In 2015 Blessing, Insko, Johnson, and Mauretour defined a $(t,r)$ broadcast dominating set, or simply a $(t,r)$ broadcast, on $G$ as a set $mathbb{T}subseteq V$ such that the sum of all signal received at each vertex $v in V$ is at least $r$. We say that $mathbb{T}$ is optimal if $|mathbb{T}|$ is minimal among all such sets $mathbb{T}$. The cardinality of an optimal $(t,r)$ broadcast on a finite graph $G$ is called the $(t,r)$ broadcast domination number of $G$. The concept of $(t,r)$ broadcast domination generalizes the classical problem of domination on graphs. In fact, the $(2,1)$ broadcasts on a graph $G$ are exactly the dominating sets of $G$. In their paper, Blessing et al. considered $(t,r)in{(2,2),(3,1),(3,2),(3,3)}$ and gave optimal $(t,r)$ broadcasts on $G_{m,n}$, the grid graph of dimension $mtimes n$, for small values of $m$ and $n$. They also provided upper bounds on the optimal $(t,r)$ broadcast numbers for grid graphs of arbitrary dimensions. In this paper, we define the density of a $(t,r)$ broadcast, which allows us to provide optimal $(t,r)$ broadcasts on the infinite grid graph for all $tgeq2$ and $r=1,2$, and bound the density of the optimal $(t,3)$ broadcast for all $tgeq2$. In addition, we give a family of counterexamples to the conjecture of Blessing et al. that the optimal $(t,r)$ and $(t+1, r+2)$ broadcasts are identical for all $tgeq1$ and $rgeq1$ on the infinite grid." @default.
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- W2772980790 date "2017-11-29" @default.
- W2772980790 modified "2023-09-27" @default.
- W2772980790 title "Optimal $(t,r)$ Broadcasts On the Infinite Grid" @default.
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