For a pair of positive integer parameters $(t,r)$, a subset $T$ of vertices of a graph $G$ is said to $(t,r)$ broadcast dominate a graph $G$ if, for any vertex $u$ in $G$, we have $\sum_{v\in T, u\in N_t(v)}(t-d(u,v))\geq r$, where where $N_{t}(v)=\{u\in V:d(u,v)<t\}$ and $d(u,v)$ denotes the distance between $u$ and $v$. This can be interpreted as each vertex $v$ of $T$ sending $\max(t-\text{d}(u,v),0)$ signal to vertices within a distance of $t-1$ away from $v$. The signal is additive and we require that every vertex of the graph receives a minimum reception $r$ from all vertices in $T$. For a finite graph the smallest cardinality among all $(t,r)$ broadcast dominating sets of a graph is called the $(t,r)$ broadcast domination number. We remark that the $(2,1)$ broadcast domination number is the domination number and the $(t,1)$ (for $t\geq 1$) is the distance domination number of a graph.
We study a family of graphs that arise as a finite subgraph of the truncated square titling, which utilizes regular squares and octagons to tile the Euclidean plane. For positive integers $m$ and $n$, we let $H_{m,n}$ be the graph consisting of $m$ rows of $n$ octagons (cycle graph on $8$ vertices). For all $t\geq 2$, we provide lower and upper bounds for the $(t,1)$ broadcast domination number for $H_{m,n}$ for all $m,n\geq 1$. We give exact $(2,1)$ broadcast domination numbers for $H_{m,n}$ when $(m,n)\in\{(1,1),(1,2),(1,3),(1,4),(2,2)\}$. We also consider the infinite truncated square tiling, denoted $H_{\infty,\infty}$, and we provide constructions of infinite $(t,r)$ broadcasts for $(t,r)\in\{(2,1),(2,2),(3,1),(3,2),(3,3),(4,1)\}$. Using these constructions we give upper bounds on the density of these broadcasts i.e., the proportion of vertices needed to $(t,r)$ broadcast dominate this infinite graph. We end with some directions for future study.
A broadcast on a graph $G=(V,E)$ is a function $f:V \rightarrow \{0,1, \ldots, \text{diam}(G)\}$ satisfying $f(v) \leq e(v)$ for all $v \in V$, where $e(v)$ denotes the eccentricity of $v$ and $\text{diam}(G)$ denotes the diameter of $G$. We say that a broadcast dominates $G$ if every vertex can hear at least one broadcasting node. The upper domination number is the maximum cost of all possible minimal broadcasts, where the cost of a broadcast is defined as $\text{cost} (f)= \sum_{v \in V}f(v)$. In this paper we establish both the upper domination number and the upper broadcast domination numb…
The domination number of a graph $G = (V,E)$ is the minimum cardinality of any subset $S \subset V$ such that every vertex in $V$ is in $S$ or adjacent to an element of $S$. Finding the domination numbers of $m$ by $n$ grids was an open problem for nearly 30 years and was finally solved in 2011 by Goncalves, Pinlou, Rao, and Thomassé. Many variants of domination number on graphs have been defined and studied, but exact values have not yet been obtained for grids. We will define a family of domination theories parameterized by pairs of positive integers $(t,r)$ where $1 \leq r \leq t$ which gen…
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 $(t,r)$ broadcast domination is a generalization of domination in which a set of broadcasting vertices emits signals of strength $t$ that decrease by 1 as they traverse each edge, and we require that every vertex in the graph receives a cumulative signal of at least $r$ from its set of broadcasting neighbors. In this paper, we extend the study of $(t,r)$ broadcast domination to directed graphs. Our…
The $(t,r)$ broadcast domination number of a graph $G$, $γ_{t,r}(G)$, is a generalization of the domination number of a graph. $γ_{t,r}(G)$ 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 do…
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 parameters $t$ and $r$. In this setting, a vertex $v \in V$ is a broadcast vertex of transmission strength $t$ if it transmits a signal of strength $t-d(u,v)$ to every vertex $u \in V$, where $d(u,v)$ denotes the distance between vertices $u$ and $v$ and $d(u,v) <t$. Given a set of broadcast vertices $S\subseteq V$, the reception at vertex $u$ is the sum of the transmissions from the broadcast vertices in…
Domination in graphs has long been studied and is applied to signal distribution problem. For example, telecommunication companies want to spread the signal from broadcast stations by transmitting varying signal strength to all receiving stations. This problem can be interpreted in a term of graph theory. Assume that broadcast companies need to spread the signal in graphs by using broadcast stations with varying signal strength so that the signal could be sent to all the vertices. However, broadcast stations with stronger signal are generally more expensive. The distribution of signal was conf…
The dual concepts of coverings and packings are well studied in graph theory. Coverings of graphs with balls of radius one and packings of vertices with pairwise distances at least two are the well-known concepts of domination and independence, respectively. In 2001, Erwin introduced \emph{broadcast domination} in graphs, a covering problem using balls of various radii, where the cost of a ball is its radius. The minimum cost of a dominating broadcast in a graph $G$ is denoted by $γ_b(G)$. The dual (in the sense of linear programming) of broadcast domination is \emph{multipacking}: a multipack…
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 $dist(T,v)$ denotes the distance between the vertices $v$ and $T$. In 2015 Blessing, Insko, Johnson, and Mauretour defined a \emph{$(t,r)$ broadcast dominating set}, or simply a \emph{$(t,r)$ broadcast}, on $G$ as a set $\mathbb{T}\subseteq V$ such that the sum of all signals received at each vertex $v \in V$ from the set of towers $\mathbb{T}$ is at least $r$. The $(t,r)$ broadcast domination numbe…
A broadcast on a nontrivial connected graph G with vertex set V is a function f from V to {0,1,...,diam(G)} such that f(v) is at most the eccentricity of v for all v in V. The weight of f is the sum of the function values taken over V. A vertex u hears f from v if f(v) is positive and d(u,v) is at most f(v). A broadcast f is dominating if every vertex of G hears f. The upper broadcast number of G is Γ_{b}(G), which is the maximum weight of a minimal dominating broadcast on G. A broadcast f is boundary independent if, for any vertex w that hears f from vertices v_{1},...,v_{k}, where k is at le…
A dominating broadcast on a graph G with vertex set V is a function f that maps V to {0,1,...,diam(G)} such that f(v) does not exceed e(v) (the eccentricity of v) for all vertices v, and each vertex u is at distance at most f(v) from a vertex v with positive f(v). The upper broadcast domination number of G is Γ_{b}(G), which equals the maximum of the sum of the function values f(v), the maximum being taken over all minimal dominating broadcasts f on G. As shown by Erwin in [D. Erwin, Cost domination in graphs, Doctoral dissertation, Western Michigan University, 2001], Γ_{b}(G) is bounded below…
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 vertex in $S$. A set $S$ satisfying these conditions is called a dominating set. In 2015 Blessing, Insko, Johnson, and Mauretour introduced $(t,r)$ broadcast domination, a generalization of graph domination parameterized by the nonnegative integers $t$ and $r$. In this setting, we say that the signal a vertex $v\in V$ receives from a tower of strength $t$ located at vertex $T$ is defined by $sig(v,T)=ma…
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 set…
Limited dominating broadcasts were proposed as a variant of dominating broadcasts, where the broadcast function is upper bounded. As a natural extension of domination, we consider dominating $2$-broadcasts along with the associated parameter, the dominating $2$-broadcast number. We prove that computing the dominating $2$-broadcast number is a NP-complete problem, but can be achieved in linear time for trees. We also give an upper bound for this parameter, that is tight for graphs as large as desired.
An integer distance digraph is the Cayley graph $Γ(\mathbb{Z},S)$ of the additive group $\mathbb{Z}$ of all integers with respect to a finite subset $S\subseteq\mathbb{Z}$. The domination ratio of $Γ(\mathbb{Z},S)$, defined as the minimum density of its dominating sets, is related to some number theory problems, such as tiling the integers and finding the maximum density of a set of integers with missing differences. We precisely determine the domination ratio of the integer distance graph $Γ(\mathbb{Z},\{1,2,\ldots,d-2,s\})$ for any integers $d$ and $s$ satisfying $d\ge2$ and $s\notin[0,d-2]$…
A dominating set $S$ in a graph $G$ is said to be perfect if every vertex of $G$ not in $S$ is adjacent to just one vertex of $S$. Given a vertex subset $S'$ of a side $P_m$ of an $m\times n$ grid graph $G$, the perfect dominating sets $S$ in $G$ with $S'=S\cap V(P_m)$ can be determined via an exhaustive algorithm $Θ$ of running time $O(2^{m+n})$. Extending $Θ$ to infinite grid graphs of width $m-1$, periodicity makes the binary decision tree of $Θ$ prunable into a finite threaded tree, a closed walk of which yields all such sets $S$. The graphs induced by the complements of such sets $S$ can …
We show there is an uncountable number of parallel total perfect codes in the integer lattice graph $Λ$ of $\R^2$. In contrast, there is just one 1-perfect code in $Λ$ and one total perfect code in $Λ$ restricting to total perfect codes of rectangular grid graphs (yielding an asymmetric, Penrose, tiling of the plane). We characterize all cycle products $C_m\times C_n$ with parallel total perfect codes, and the $d$-perfect and total perfect code partitions of $Λ$ and $C_m\times C_n$, the former having as quotient graph the undirected Cayley graphs of $\Z_{2d^2+2d+1}$ with generator set $\{1,2d^…
A broadcast on a connected graph $G=(V,E)$ is a function $f:V\rightarrow \{0,1,\dots,\operatorname{diam}(G)\}$ such that $f(v)\leq e(v)$ (the eccentricity of $v$) for all $v\in V$ if $|V|\geq2$, and $f(v)=1$ if $V=\{v\}$. The cost of $f$ is $σ(f)=\sum_{v\in V}f(v)$. Let $V_{f}% ^{+}$ denote the set of vertices $v$ such that $f(v)$ is positive. A vertex $u$ hears $f$ from $v\in V_{f}^{+}$ if the distance $d(u,v)\leq f(v)$. When $f$ is a broadcast such that every vertex $x$ that hears $f$ from more than one vertex in $V_{f}^{+}$ also satisfies $d(x,u)\geq f(u)$ for all $u\in V_{f}^{+}$, we say t…
A broadcast on a graph $G=(V,E)$ is a function $f: V\longrightarrow \{0,\ldots,\operatorname{diam}(G)\}$ such that $f(v)\leq e\_G(v)$ for every vertex $v\in V$, where$\operatorname{diam}(G)$ denotes the diameter of $G$ and $e\_G(v)$ the eccentricity of $v$ in $G$. The cost of such a broadcast is then the value $\sum\_{v\in V}f(v)$.Various types of broadcast functions on graphs have been considered in the literature, in relation with domination, irredundence, independenceor packing, leading to the introduction of several broadcast numbers on graphs.In this paper, we determine these broadcast nu…
A broadcast on a nontrivial connected graph G is a function f from the vertices of G to the non-negative integers such that f(v) does not exceed e(v) (the eccentricity of v) for each vertex v. If G is disconnected, we define a broadcast on G as the union of broadcasts on its components. In a search for the best way to generalise the concept of independent sets in graphs to independent broadcasts, there are several ways to look at an independent set X of a graph G. One way is from the point of view of the vertices in X: no two vertices are adjacent -- the usual definition. Another way is from t…
A vertex subset M of a graph G is a multipacking if for each vertex v, and each positive integer s less than or equal to the diameter of G, v is within distance s of at most s vertices of M. The multipacking number of a graph is the maximum cardinality of a multipacking of G. A generalization of 2-packings, multipackings offer interesting insight into the minimum cost broadcast domination problem. This paper surveys recent results in the study of multipackings, including the equality of the multipacking number and broadcast number in trees, an extension of Farber's Algorithm for finding domina…
