Related papers: On the integer {k}-domination number of circulant …

A disjunctive dominating set of a graph $G$ is a set $D \subseteq V(G)$ such that every vertex in $V(G)\setminus D$ has a neighbor in $D$ or has at least two vertices in $D$ at distance $2$ from it. The disjunctive domination number of $G$,…

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 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 $G$ 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…

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…

Let $k$ be a positive integer and let $G$ be a graph with vertex set $V(G)$. A subset $D \subseteq V(G)$ is a $k$-dominating set if every vertex outside $D$ is adjacent to at least $k$ vertices in $D$. The $k$-domination number…

A set $D \subseteq V(G)$ is a \emph{dominating set} of a graph $G$ if every vertex of $G$ not in $D$ is adjacent to at least one vertex in $D$. A \emph{minimum dominating set} of $G$, also called a $\gamma(G)$-set, is a dominating set of…

Given a graph $G$, the $k$-dominating graph of $G$, $D_k(G)$, is defined to be the graph whose vertices correspond to the dominating sets of $G$ that have cardinality at most $k$. Two vertices in $D_k(G)$ are adjacent if and only if the…

A set $D$ of vertices of a graph $G$ is a dominating set of $G$ if every vertex in $V_G-D$ is adjacent to at least one vertex in $D$. The domination number (upper domination number, respectively) of a graph $G$, denoted by $\gamma(G)$…

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…

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…

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 finite undirected graph. A set $S$ of vertices in $V$ is said to be total $k$-dominating if every vertex in $V$ is adjacent to at least $k$ vertices in $S$. The total $k$-domination number, $\gamma_{kt}(G)$, is the…

Let $G_1$ and $G_2$ be disjoint copies of a graph $G$, and let $f: V(G_1) \rightarrow V(G_2)$ be a function. Then a \emph{functigraph} $C(G, f)=(V, E)$ has the vertex set $V=V(G_1) \cup V(G_2)$ and the edge set $E=E(G_1) \cup E(G_2) \cup…

Let ${[n] \choose k}$ and ${[n] \choose l}$ $( k > l ) $ where $[n] = \{1,2,3,...,n\}$ denote the family of all $k$-element subsets and $l$-element subsets of $[n]$ respectively. Define a bipartite graph $G_{k,l} = ({[n] \choose k},{[n]…

For any graph G = (V, E) and proportion $p\in(0,1]$, a set $S\subseteq V$ is a p-dominating set if $\frac{|N[S]|}{|V|}\geq p$. The $p$-domination number $\gamma_{p}(G)$ equals the minimum cardinality of a $p$-dominating set in G. For a…

A subset $D$ of vertices of a graph $G$ is a total dominating set if every vertex of $G$ is adjacent to at least one vertex of $D$. The total dominating set $D$ is called a total co-independent dominating set if the subgraph induced by…

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…

A set $D$ of vertices in $G$ is a disjunctive dominating set in $G$ if every vertex not in $D$ is adjacent to a vertex of $D$ or has at least two vertices in $D$ at distance $2$ from it in $G$. The disjunctive domination number,…

The modular product $G\diamond H$ of graphs $G$ and $H$ is a graph on vertex set $V(G)\times V(H)$. Two vertices $(g,h)$ and $(g^{\prime},h^{\prime})$ of $G\diamond H$ are adjacent if $g=g^{\prime}$ and $hh^{\prime}\in E(H)$, or…