Related papers: Edge Roman domination on graphs

Given a graph $G=(V,E)$, $f:V \rightarrow \{0,1,2 \}$ is the Italian dominating function of $G$ if $f$ satisfies $\sum_{u \in N(v)}f(u) \geq 2$ when $f(v)=0$. Denote $w(f)=\sum_{v \in V}f(v)$ as the weight of $f$. Let…

Given a graph $G$, a dominating set of $G$ is a set $S$ of vertices such that each vertex not in $S$ has a neighbor in $S$. The domination number of $G$, denoted $\gamma(G)$, is the minimum size of a dominating set of $G$. The independent…

A set $S$ of vertices in a graph $G$ is a dominating set of $G$ if every vertex not in $S$ is adjacent to a vertex in~$S$. An independent dominating set in $G$ is a dominating set of $G$ with the additional property that it is an…

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 $\{0,1,\dots, t\}$ be abbreviated by $[t].$ A double Roman dominating function (DRDF) on a graph $\Gamma=(V,E)$ is a map $l:V\rightarrow [3]$ satisfying \textrm{(i)} if $l(r)=0$ then there must be at least two neighbors labeled 2 under…

Since Reed conjectured in 1996 that the domination number of a connected cubic graph of order $n$ is at most $\lceil \frac13 n \rceil$, the domination number of cubic graphs has been extensively studied. It is now known that the conjecture…

Let $G$ be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, $\gamma_t(G)$. A set $S$ of vertices in $G$ is a…

A subset $M$ of the edges of a graph $G$ is a matching if no two edges in $M$ are incident. A maximal matching is a matching that is not contained in a larger matching. A subset $S$ of vertices of a graph $G$ with no isolated vertices is a…

An edge dominating set $F$ of a graph $G=(V,E)$ is an \textit{edge cut dominating set} if the subgraph $\langle V,G-F \rangle$ is disconnected. The \textit{edge cut domination number} $\gamma_{ct}(G)$ of $G$ is the minimum cardinality of an…

Roman domination is one of the many variants of domination that keeps most of the complexity features of the classical domination problem. We prove that Roman domination behaves differently in two aspects: enumeration and extension. We…

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…

Given a graph $G$, a dominating set of $G$ is a set $S$ of vertices such that each vertex not in $S$ has a neighbor in $S$. The domination number of $G$, denoted $\gamma(G)$, is the minimum size of a dominating set of $G$. The independent…

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…

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…

A vertex set $S$ of a graph $G$ is a \emph{dominating set} if each vertex of $G$ either belongs to $S$ or is adjacent to a vertex in $S$. The \emph{domination number} $\gamma(G)$ of $G$ is the minimum cardinality of $S$ as $S$ varies over…

A graph $G$ is a \emph{cover} of a graph $F$ if there exists an onto mapping $\pi : V(G) \to V(F)$, called a (\emph{covering}) \emph{projection}, such that $\pi$ maps the neighbours of any vertex $v$ in $G$ bijectively onto the neighbours…

In this paper, an upper bound for the perfect Italian domination number of the cartesian product of any two graphs is obtained and the exact value of this parameter for cartesian product of some special graphs are obtained. We have also…

Given a graph $G$ with vertex set $V$, $f : V \rightarrow \{0, 1, 2\}$ is a \emph{Roman $\{2\}$-dominating function} (or \emph{italian dominating function}) of $G$ if for every vertex $v\in V$ with $f(v) =0$, either there exists a vertex…

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 simple undirected graph. $G$ is a circulant graph defined on $V=\mathbb{Z}_n$ with difference set $D\subseteq \{1,2,\ldots,\lfloor\frac{n}{2}\rfloor\}$ provided two vertices $i$ and $j$ in $\mathbb{Z}_n$ are adjacent if…