Related papers: Efficient $k$-limited Dominating Broadcasts in Pro…

Given a graph $G$, a subset $S$ of vertices of $G$ is an efficient dominating set ($EDS$) if $|N[v] \cap S|=1,$ for all $v\in V(G)$. A graph $G$ is efficiently dominatable if it possesses an $EDS$. The efficient domination number of G is…

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…

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…

A vertex set $D$ in a finite undirected graph $G$ is an efficient dominating set (e.d.s. for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The Efficient Domination (ED) problem, which asks for the existence…

Given a graph $G = (V,E)$, a \emph{perfect dominating set} is a subset of vertices $V' \subseteq V(G)$ such that each vertex $v \in V(G)\setminus V'$ is dominated by exactly one vertex $v' \in V'$. An \emph{efficient dominating set} is a…

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…

A dominating set $S$ is an Isolate Dominating Set (IDS) if the induced subgraph $G[S]$ has at least one isolated vertex. In this paper, we initiate the study of new domination parameter called, isolate secure domination. An isolate…

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…

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…

For a graph $G$ let $\gamma (G)$ be its domination number. We define a graph G to be (i) a hypo-efficient domination graph (or a hypo-$\mathcal{ED}$ graph) if $G$ has no efficient dominating set (EDS) but every graph formed by removing a…

Given an undirected graph $G=(V,E)$, a vertex $v\in V$ is edge-vertex (ev) dominated by an edge $e\in E$ if $v$ is either incident to $e$ or incident to an adjacent edge of $e$. A set $S^{ev}\subseteq E$ is an edge-vertex dominating set…

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is…

Dominating broadcasting is a domination-type structure that models a transmission antenna network. In this paper, we study a limited version of this structure, that was proposed as a common framework for both broadcast and classical…

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…

Let $k$ be a positive integer and $G=(V,E)$ be a graph of minimum degree at least $k-1$. A function $f:V\rightarrow \{-1,1\}$ is called a \emph{signed $k$-dominating function} of $G$ if $\sum_{u\in N_G[v]}f(u)\geq k$ for all $v\in V$. The…

Let $G=(V,E)$ be a simple undirected graph. The open neighbourhood of a vertex $v$ in $G$ is defined as $N_G(v)=\{u\in V~|~ uv\in E\}$; whereas the closed neighbourhood is defined as $N_G[v]= N_G(v)\cup \{v\}$. For an integer $k$, a subset…

For a graph $ G = (V, E) $ with a vertex set $ V $ and an edge set $ E $, a function $ f : V \rightarrow \{0, 1, 2, . . . , diam(G)\} $ is called a \emph{broadcast} on $ G $. For each vertex $ u \in V $, if there exists a vertex $ v $ in $…

Given a graph $G = (V, E)$, a set $S \subseteq V \cup E$ of vertices and edges is called a mixed dominating set if every vertex and edge that is not included in $S$ happens to be adjacent or incident to a member of $S$. The mixed domination…

A set $D \subseteq V(G)$ is a \emph{total dominating set} of $G$ if for every vertex $v \in V(G)$ there exists a vertex $u \in D$ such that $u$ and $v$ are adjacent. A total dominating set of $G$ of minimum cardinality is called a…