Related papers: Domination number of modular product graphs

For a graph $G$, the $\gamma$-graph of $G$, $G(\gamma)$, is the graph whose vertices correspond to the minimum dominating sets of $G$, and where two vertices of $G(\gamma)$ are adjacent if and only if their corresponding dominating sets in…

In a graph G, a vertex dominates itself and its neighbors. A subset S of V is called a dominating set in G if every vertex in V is dominated by at least one vertex in S. The domination number gamma G is the minimum cardinality of a…

Let $G=(V, E)$ be a graph. A set $S\subseteq V(G)$ is a {\it dominating set} of $G$ if every vertex in $V\setminus S$ is adjacent to a vertex of $S$. The {\it domination number} of $G$, denoted by $\gamma(G)$, is the cardinality of a…

A dominating set $S$ of a graph $G$ of order $n$ is a subset of the vertices of $G$ such that every vertex is either in $S$ or adjacent to a vertex of $S$. %The domination number $G$, denoted $\gamma (G)$, is the cardinality of the smallest…

Let $G=(V(G),E(G))$ be a simple connected and undirected graph with vertex set $V(G)$ and edge set $E(G)$. A set $S \subseteq V(G)$ is a $dominating$ $set$ if for each $v \in V(G)$ either $v \in S$ or $v$ is adjacent to some $w \in S$. That…

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…

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…

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 $G=(V,E)$ 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 dominating (respectively, total dominating) set $S$ of a digraph $D$ is a set of vertices in $D$ such that the union of the closed (respectively, open) out-neighborhoods of vertices in $S$ equals the vertex set of $D$. The minimum size of…

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…

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 $G=(V,E)$ 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…

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 vertex set $D$ of a graph $G$ is said to be a dominating set if every vertex of $V(G)\setminus D$ is adjacent to at least a vertex in $D$, and the domination number $\gamma(G)$ ($\gamma$, for short) is the minimum cardinality of all…

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…

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 $S$ of vertices of a graph $G$ is a dominating set in $G$ if every vertex outside of $S$ is adjacent to at least one vertex belonging to $S$. A domination parameter of $G$ is related to those sets of vertices of a graph satisfying…

A longest sequence $(v_1,\ldots,v_k)$ of vertices of a graph $G$ is a Grundy total dominating sequence of $G$ if for all $i$, $N(v_i) \setminus \bigcup_{j=1}^{i-1}N(v_j)\not=\emptyset$. The length $k$ of the sequence is called the Grundy…

As a natural variant of domination in graphs, Dankelmann et al. [Domination with exponential decay, Discrete Math. 309 (2009) 5877-5883] introduce exponential domination, where vertices are considered to have some dominating power that…