Related papers: On the total and strong version for Roman dominati…

We continue the study of restrained double Roman domination in graphs. For a graph $G=\big{(}V(G),E(G)\big{)}$, a double Roman dominating function $f$ is called a restrained double Roman dominating function (RDRD function) if the subgraph…

Let $k$ be a positive integer. A {\em Roman $k$-dominating function} on a graph $G$ is a labeling $f:V (G)\longrightarrow \{0, 1, 2\}$ such that every vertex with label 0 has at least $k$ neighbors with label 2. A set…

In this paper, we define a new domination-like invariant of graphs. Let $\mathbb{R}^{+}$ be the set of non-negative numbers. Let $c\in \mathbb{R}^{+}-\{0\}$ be a number, and let $G$ be a graph. A function $f:V(G)\rightarrow \mathbb{R}^{+}$…

Let $G=(V(G),E(G))$ be a simple graph. A restrained double Roman dominating function (RDRD-function) of $G$ is a function $f: V(G) \rightarrow \{0,1,2,3\}$ satisfying the following properties: if $f(v)=0$, then the vertex $v$ has at least…

Given a graph $G=(V,E)$, a function $f:V\to \{0,1,2\}$ is said to be a \emph{Roman Dominating function} (RDF) if for every $v\in V$ with $f(v)=0$, there exists a vertex $u\in N(v)$ such that $f(u)=2$. A Roman Dominating function $f$ is said…

Consider a graph $G = (V, E)$ and a function $f: V \rightarrow \{0, 1, 2\}$. A vertex $u$ with $f(u)=0$ is defined as \emph{undefended} by $f$ if it lacks adjacency to any vertex with a positive $f$-value. The function $f$ is said to be a…

An edge Roman dominating function of a graph $G$ is a function $f\colon E(G) \rightarrow \{0,1,2\}$ satisfying the condition that every edge $e$ with $f(e)=0$ is adjacent to some edge $e'$ with $f(e')=2$. The edge Roman domination number of…

Let $G=(V, E)$ be a simple undirected graph with no isolated vertex. A set $D_t\subseteq V$ is a total dominating set of $G$ if $(i)$ $D_t$ is a dominating set, and $(ii)$ the set $D_t$ induces a subgraph with no isolated vertex. The total…

A numbering $f$ of a graph $G$ of order $n$ is a labeling that assigns distinct elements of the set $\left\{ 1,2,\ldots ,n\right\} $ to the vertices of $G$. The strength $\textrm{str}_{f}\left( G\right)$ of a numbering $f:V\left( G\right)…

The Roman domination in a graph $G$ is a variant of the classical domination, defined by means of a so-called Roman domination function $f\colon V(G)\to \{0,1,2\}$ such that if $f(v)=0$ then, the vertex $v$ is adjacent to at least one…

For a graph $G=(V,E)$, a double roman dominating function (DRDF) is a function $f : V \longrightarrow \{0, 1, 2,3\}$ having the property that if $f(v)=0$ for some vertex $v$, then $v$ has at least two neighbors assigned $2$ under $f$ or one…

For a graph $G=(V,E)$, a Roman $\{2\}$-dominating function (R2DF)$f:V\rightarrow \{0,1,2\}$ has the property that for every vertex $v\in V$ with $f(v)=0$, either there exists a neighbor $u\in N(v)$, with $f(u)=2$, or at least two neighbors…

A set $S$ of vertices in a graph $G(V,E)$ is called a dominating set if every vertex $v\in V$ is either an element of $S$ or is adjacent to an element of $S$. A set $S$ of vertices in a graph $G(V,E)$ is called a total dominating set if…

A Roman $\{3\}$-dominating function on a graph $G = (V, E)$ is a function $f: V \rightarrow \{0, 1, 2, 3\}$ such that for each vertex $u \in V$, if $f(u) = 0$ then $\sum_{v \in N(u)} f(v) \geq 3$ and if $f(u) = 1$ then $\sum_{v \in N(u)}…

Given a graph $G$ with vertex set $V(G)$, a mapping $h : V(G) \rightarrow \lbrace 0, 1, 2, 3, 4, 5 \rbrace$ is called a quadruple Roman dominating function (4RDF) for $G$ if it holds the following. Every vertex $x$ such that $h(x)\in…

Given a graph $G=(V,E)$, a function $f:V\to \{0,1,2\}$ is said to be a \emph{Roman Dominating function} if for every $v\in V$ with $f(v)=0$, there exists a vertex $u\in N(v)$ such that $f(u)=2$. A Roman Dominating function $f$ is said to be…

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 any integer $k\geq 1$ and any graph $G=(V,E)$ with minimum degree at least $k-1$, we define a function $f:V\rightarrow \{0,1,2\}$ as a Roman $k$-tuple dominating function on $G$ if for any vertex $v$ with $f(v)=0$ there exist at least…

A set $D$ of vertices of a simple graph $G=(V,E)$ is a strong dominating set, if for every vertex $x\in \overline{D}=V\setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $deg(x)\leq deg(y)$. The strong domination number…

In a directed graph $D$, a vertex subset $S\subseteq V$ is a total dominating set if every vertex of $D$ has an in-neighbor from $S$. A total dominating set exists if and only if every vertex has at least one in-neighbor. We call the…