Related papers: On the outer independent total double Roman domina…

In the Roman domination problem, an undirected simple graph $G(V,E)$ is given. The objective of Roman domination problem is to find a function $f:V\rightarrow {\{0,1,2\}}$ such that for any vertex $v\in V$ with $f(v)=0$ must be adjacent to…

A set $S$ of vertices of a graph $G$ is a dominating set for $G$ if every vertex outside of $S$ is adjacent to at least one vertex belonging to $S$. The minimum cardinality of a dominating set for $G$ is called the domination number of $G$.…

Let $\gamma(G)$ denote the domination number of a graph $G$. A {\it Roman domination function} of a graph $G$ is a function $f: V\to\{0,1,2\}$ such that every vertex with 0 has a neighbor with 2. The {\it Roman domination number}…

In a graph $G$, a vertex dominates itself and its neighbors. A subset $S\subseteq V(G)$ is said to be a double dominating set of $G$ if $S$ dominates every vertex of $G$ at least twice. The double domination number $\gamma_{\times 2}(G)$ is…

Let $G=(V,E)$ be a graph. A subset $D$ of $V$ is a \textit{restrained dominating set} if every vertex in $V \setminus D$ is adjacent to a vertex in $D$ and to a vertex in $V \setminus D$. The \textit{restrained domination number}, denoted…

We study a variant of domination, called Roman domination, where we must assign to each vertex one of the labels 0, 1, or 2 and require that every vertex with label 0 has a neighbour with label 2. We study the problem of finding a low-cost…

Given a graph $G = (V, E)$, a signed Roman dominating function is a function $f: V \rightarrow \{-1, 1, 2\}$ such that for every vertex $u \in V$: $\sum_{v \in N[u]} f(v) \geq 1$ and for every vertex $u \in V$ with $f(u) = -1$, there exists…

An Italian dominating function (IDF) of a graph G is a function $ f: V(G) \rightarrow \{0,1,2\} $ satisfying the condition that for every $ v\in V $ with $ f(v) = 0$, $\sum_{ u\in N(v)} f(u) \geq 2. $ The weight of an IDF on $G$ is the sum…

The {\em independent domination number} $\gamma^i(G)$ of a graph $G$ is the maximum, over all independent sets $I$, of the minimal number of vertices needed to dominate $I$. It is known \cite{abz} that in chordal graphs $\gamma^i$ is equal…

At least two different notions have been published under the name "majority domination in graphs": Majority dominating functions and majority dominating sets. In this work we extend the former concept to digraphs. Given a digraph $D=(V,A),$…

An \textit{Italian dominating function} on a digraph $D$ with vertex set $V(D)$ is defined as a function $f : V(D) \rightarrow \{0, 1, 2\}$ such that every vertex $v \in V(D)$ with $f(v) = 0$ has at least two in-neighbors assigned $1$ under…

Let $G$ be a nontrivial connected graph with vertex set $V(G)$. A set of vertices $D\subseteq V(G)$ is called a total outer-independent dominating set of $G$ if every vertex of $G$ is adjacent to at least one vertex in $D$, and…

In a graph $G=(V,E)$, each vertex $v\in V$ is labelled with $0$, $1$ or $2$ such that each vertex labelled with $0$ is adjacent to at least one vertex labelled $2$ or two vertices labelled $1$. Such kind of labelling is called an Italian…

Let $k$ be a positive integer. A $k$-rainbow domination function (kRDF) of a graph $G$ is a function $f$ from $V(G)$ to the set of all subsets of $\{1,2,\dots,k\}$ such that every vertex $v \in V(G)$ with $f(v) = \emptyset$ satisfies…

For a function $f : V(G ) \rightarrow \{0, 1, 2\}$ we denote by $V_i$ the set of vertices to which the value $i$ is assigned by $f$, i.e. $V_i = \{ x \in V (G ) : f(x ) = i \}$. If a function $f: V(G) \rightarrow \{0,1,2\}$ satisfying the…

Let $G$ be a graph with vertex set $V(G)$. A function $f:V(G)\rightarrow \{0,1,2\}$ is a Roman dominating function on $G$ if every vertex $v\in V(G)$ for which $f(v)=0$ is adjacent to at least one vertex $u\in V(G)$ such that $f(u)=2$. The…

For a positive integer $k$, a $\{k\}$-Roman dominating function of a graph $G = (V,E)$ is a function $f\colon V \rightarrow \{0,1,\ldots,k\}$ satisfying $f (N(v)) \geq k$ for each vertex $v\in V$ with $f (v) = 0$. Every graph $G$ satisfies…

Given a simple graph $G$, a dominating set in $G$ is a set of vertices $S$ such that every vertex not in $S$ has a neighbor in $S$. Denote the domination number, which is the size of any minimum dominating set of $G$, by $\gamma(G)$. For…

For a graph $G=(V(G),E(G))$, an Italian dominating function (ID function) $f:V(G)\rightarrow\{0,1,2\}$ has the property that for every vertex $v\in V(G)$ with $f(v)=0$, either $v$ is adjacent to a vertex assigned $2$ under $f$ or $v$ is…

A locating-dominating set of a graph $G$ is a dominating set $D$ of $G$ with the additional property that every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N(u)…