Related papers: Critical properties on Roman domination graphs

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…

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…

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…

For $c\in \mathbb{R}^{+}\cup \{\infty \}$ and a graph $G$, a function $f:V(G)\rightarrow \{0,1,c\}$ is called a $c$-self dominating function of $G$ if for every vertex $u\in V(G)$, $f(u)\geq c$ or $\max\{f(v):v\in N_{G}(u)\}\geq 1$ where…

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…

Given a graph $G$ with vertex set $V$, an outer independent Roman dominating function (OIRDF) is a function $f$ from $V(G)$ to $\{0, 1, 2\}$ for which every vertex with label $0$ under $f$ is adjacent to at least a vertex with label $2$ but…

Given a graph $G=(V,E)$, a function $f:V\rightarrow \{0,1,2,3\}$ having the property that if $f(v)=0$, then there exist $ v_{1},v_{2}\in N(v)$ such that $f(v_{1})=f(v_{2})=2$ or there exists $ w \in N(v)$ such that $f(w)=3$, and if…

Let $G$ be a connected graph of order $n$ with vertex set $V(G)$. A subset $S\subseteq V(G)$ is an $(a,b)$-dominating set if every vertex $v\in S$ is adjacent to at least $a$ vertices in $S$ and every $v\in V\setminus S$ is adjacent to at…

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

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…

Let $G=(V,E)$ be a graph of order $n$ and let $\gamma _{R}(G)$ and $\partial (G)$ denote the Roman domination number and the differential of $G,$ respectively. In this paper we prove that for any integer $k\geq 0$, if $G$ is a graph of…

An independent double Roman dominating function (IDRDF) on a graph $G=(V,E)$ is a function $f:V(G)\rightarrow \{0,1,2,3\}$ having the property that if $f(v)=0$, then the vertex $v$ has at least two neighbors assigned $2$ under $f$ or one…

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…

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…

A set $S\subseteq V$ is a dominating set of $G$ if every vertex in $V - S$ is adjacent to at least one vertex in $S$. The domination number $\gamma(G)$ of $G$ equals the minimum cardinality of a dominating set $S$ in $G$; we say that such a…

Let $\gamma(G)$ be the domination number of a graph $G$. A graph $G$ is \emph{domination-vertex-critical}, or \emph{$\gamma$-vertex-critical}, if $\gamma(G-v)< \gamma(G)$ for every vertex $v \in V(G)$. In this paper, we show that: Let $G$…

Given a graph G, the domination number gamma(G) of G is the minimum order of a set S of vertices such that each vertex not in S is adjacent to some vertex in S. Equivalently, label the vertices from {0, 1} so that the sum over each closed…

A subset $S\subseteq V$ in a graph $G=(V,E)$ is a total $[1,2]$-set if, for every vertex $v\in V$, $1\leq |N(v)\cap S|\leq 2$. The minimum cardinality of a total $[1,2]$-set of $G$ is called the total $[1,2]$-domination number, denoted by…

For a graph $G$, let $\gamma_R(G)$ and $\gamma_{r2}(G)$ denote the Roman domination number of $G$ and the $2$-rainbow domination number of $G$, respectively. It is known that $\gamma_{r2}(G)\leq \gamma_R(G)\leq \frac{3}{2}\gamma_{r2}(G)$.…

The 2-domination number $\gamma_2(G)$ of a graph $G$ is the minimum cardinality of a set $ D \subseteq V(G) $ for which every vertex outside $ D $ is adjacent to at least two vertices in $ D $. Clearly, $ \gamma_2(G) $ cannot be smaller…