Related papers: Roman domination in weighted graphs

Let $G=(V,E)$ be a graph. A subset $D\subseteq V$ is a dominating set if every vertex not in $D$ is adjacent to a vertex in $D$. The domination number of $G$, denoted by $\gamma(G)$, is the smallest cardinality of a dominating set of $G$.…

A graph $G=(V,E)$ is called equidominating if there exists a value $t \in \mathbb{N}$ and a weight function $\omega : V \rightarrow \mathbb{N}$ such that the total weight of a subset $D\subseteq V$ is equal to $t$ if and only if $D$ is a…

A signed dominating function of graph $\Gamma$ is a function $g :V(\Gamma) \longrightarrow \{-1,1\}$ such that $\sum_{u \in N[v]}g(u) >0$ for each $v \in V(\Gamma)$. The signed domination number $\gamma_{_S}(\Gamma)$ is the minimum weight…

This work is related to the extension of the well-known problem of Roman domination in graph theory to fuzzy graphs. A variety of approaches have been used to explore the concept of domination in fuzzy graphs. This study uses the concept of…

Let $G$ be a graph. A set $S$ of vertices in $G$ dominates the graph if every vertex of $G$ is either in $S$ or a neighbor of a vertex in $S$. Finding a minimal cardinality set which dominates the graph is an NP-complete problem. The graph…

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

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…

Let $G=(V,E)$ be a graph. For some $\alpha$ with $0<\alpha \leq 1$, a subset $S$ of $V$ is said to be a $\alpha$-partial dominating set if $|N[S]|\geq \alpha |V|$. The size of a smallest such $S$ is called the $\alpha$-partial domination…

While a number of bounds are known on the zero forcing number $Z(G)$ of a graph $G$ expressed in terms of the order of a graph and maximum or minimum degree, we present two bounds that are related to the (upper) total domination number…

A set $D$ of vertices of a graph $G$ is a dominating set if each vertex of $V(G)\setminus D$ is adjacent to some vertex of $D$. The domination number of $G$, $\gamma(G)$, is the minimum cardinality of a dominating set of $G$. A graph $G$ is…

Given a graph G equals (V,E), a subset S subset of V is a dominating set if every vertex in V minus S is adjacent to some vertex in S. The dominating set with the least cardinality, gamma, is called a gamma-set which is commonly known as a…

A non-empty set $S\subseteq V (G)$ of the simple graph $G=(V(G),E(G))$ is an independent dominating set of $G$ if every vertex not in $S$ is adjacent with some vertex in $S$ and the vertices of $S$ are pairwise non-adjacent. The independent…

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…

We propose the conjecture that the domination number $\gamma(G)$ of a $\Delta$-regular graph $G$ with $\Delta\geq 1$ is always at most its edge domination number $\gamma_e(G)$, which coincides with the domination number of its line graph.…

A dominating set of a graph $G$ is a set $D\subseteq V(G)$ such that \-every vertex of $G$ is either in $D$ or is adjacent to a vertex in $D$. The domination number of $G$, $\gamma(G)$, is the minimum order of a dominating set. A subset $R$…

Roman domination and its higher-order extensions have attracted considerable attention due to their natural interpretation in terms of defensive resource allocation on networks. The recently introduced $[k]$-Roman domination framework…

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

A dominating set of a graph $G$ is a set $D\subseteq V_G$ such that every vertex in $V_G-D$ is adjacent to at least one vertex in $D$, and the domination number $\gamma(G)$ of $G$ is the minimum cardinality of a dominating set of $G$. In…

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…

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…