Optimal (t, r) broadcasts on the infinite grid

  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.},
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
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.
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
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.


Radial trees
Broadcast Domination in Tori
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
Broadcasts in graphs
New Upper Bounds on the Distance Domination Numbers of Grids
In his 1992 Ph.D. thesis Chang identified an efficient way to dominate $m \times n$ grid graphs and conjectured that his construction gives the most efficient dominating sets for relatively large
On the r-domination number of a graph
R-Domination in Graphs
A linear algorithm to find a minimum 1-basis (a minimum dominating set) when G is a tree and a linear algorithm that solves the problem for any forest is presented.
Broadcast domination in graph products of paths
The value of γb(G), whenever G is the strong product, the direct product and the lexicographic product of two paths, is evaluated.
Distributed dominating sets on grids
This paper presents a distributed algorithm for finding near optimal dominating sets on grids that is generalized for the k-distance dominating set problem, where all grid vertices are within distance k of the vertices in the dominating set.
The domination numbers of the 5 × n and 6 × n grid graphs
The k × n grid graph is the product Pk × Pn of a path of length k − 1 and a route of length n − 1, and formulas found by E. O. Hare for the domination numbers of P5 → Pn and P6 × PN are proved.