Optimal (t, r) broadcasts on the infinite grid

@article{Drews2019OptimalR,
  title={Optimal (t, r) broadcasts on the infinite grid},
  author={Benjamin F. Drews and Pamela E. Harris and Timothy W. Randolph},
  journal={Discret. Appl. Math.},
  year={2019},
  volume={255},
  pages={183-197}
}
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10 Citations
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2
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10 Citations

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Citation Type

Asymptotically Optimal Bounds for (t,2) Broadcast Domination on Finite Grids
Let $G=(V,E)$ be a graph and $t,r$ be positive integers. The \emph{signal} that a tower vertex $T$ of signal strength $t$ supplies to a vertex $v$ is defined as $sig(T,v)=max(t-dist(T,v),0),$ where…
Computing upper bounds for optimal density of $(t,r)$ broadcasts on the infinite grid
The domination number of a finite graph $G$ with vertex set $V$ is the cardinality of the smallest set $S\subseteq V$ such that for every vertex $v\in V$ either $v\in S$ or $v$ is adjacent to a…
(t,r) broadcast domination in the infinite grid
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…
Broadcast Domination of Triangular Matchstick Graphs and the Triangular Lattice
Blessing, Insko, Johnson and Mauretour gave a generalization of the domination number of a graph $G=(V,E)$ called the $(t,r)$ broadcast domination number which depends on the positive integer…
Bounds On $(t,r)$ Broadcast Domination of $n$-Dimensional Grids
TLDR
Some upper and lower bounds on the density of a dominating pattern of an infinite grid, as well as methods of computing them are described.
Efficient (t, r) broadcast dominating sets of the triangular lattice
The general position number of integer lattices
On $(t,r)$ broadcast domination of directed graphs
A dominating set of a graph G is a set of vertices that contains at least one endpoint of every edge on the graph. The domination number of G is the order of a minimum dominating set of G. The (t, r)…
Projects in (t, r) Broadcast Domination
TLDR
This study focuses on (t, r) broadcast domination, a variant with a connection to the placement of cellphone towers, where some vertices send out a signal to nearby vertices and where all vertices must receive a minimum predetermined amount of this signal.
The (t, r) broadcast domination number of some regular graphs

References

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A broadcast on a graph G is a function f : V (G) ! f0;1;:::;diam(G)g such that for every vertex v 2 V (G), f(v) e(v), where diam(G) is the diameter of G, and e(v) is the eccentricity of v. In…
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