Related papers: General bounds on limited broadcast domination

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…

For an integer $k \ge 1$, a (distance) $k$-dominating set of a connected graph $G$ is a set $S$ of vertices of $G$ such that every vertex of $V(G) \setminus S$ is at distance at most~$k$ from some vertex of $S$. The $k$-domination number,…

For a graph $G$, a function $f:V(G) \to \{0,1,2\}$ is called a $2$-limited dominating broadcast on $G$ if for every vertex $u$, there exists a vertex $v$ such that $f(v)>0$ and the distance between $u$ and $v$ in $G$ is at most $f(v)$. The…

A broadcast on a nontrivial connected graph G=(V,E) is a function f from V(G) to {0,1,...,diam(G)} such that f(v) does not exceed the eccentricity of v. The cost of f is the sum of the function values. A broadcast f is dominating if each…

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 $G$.…

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…

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…

In this paper, we study a variant of graph domination known as $(t, r)$ broadcast domination, first defined in Blessing, Insko, Johnson, and Mauretour in 2015. In this variant, each broadcast provides $t-d$ reception to each vertex a…

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…

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…

For $k \geq 1$ and a graph $G$ without isolated vertices, a \emph{total (distance) $k$-dominating set} of $G$ is a set of vertices $S \subseteq V(G)$ such that every vertex in $G$ is within distance $k$ to some vertex of $S$ other than…

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…

In 2001, Erwin introduced broadcast domination in graphs. It is a variant of classical domination where selected vertices may have different domination powers. The minimum cost of a dominating broadcast in a graph $G$ is denoted…

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…

We establish upper and lower bounds for the 2-limited broadcast domination number of various grid graphs, in particular the Cartesian product of two paths, a path and a cycle, and two cycles. The upper bounds are derived by explicit…

In a graph $ G $, a subset of vertices $ S $ is called an efficient dominating set (EDS) if every vertex in the graph is uniquely dominated by exactly one vertex in $ S $. A graph is said to be efficiently dominatable if it contains an EDS.…

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…

We first introduce the concept of (k,k',k'')-domination numbers in graphs, which is a generalization of many domination parameters. Then we find lower and upper bounds for this parameter, which improve many well-known results in…

Let $G=( V(G), E(G) )$ be a connected graph with vertex set $V(G)$ and edge set $E(G)$. We say a subset $D$ of $V(G)$ dominates $G$ if every vertex in $V \setminus D$ is adjacent to a vertex in $D$. A generalization of this concept is…

The notion of power domination arises in the context of monitoring an electric power system with as few phase measurement units as possible. The $k-$power domination number of a graph $G$ is the minimum cardinality of a $k-$power dominating…