Related papers: Open k-monopolies in graphs: complexity and relate…

Given a simple graph $G$, a dominating set in $G$ is a set of vertices $S$ such that every vertex not in $S$ has a neighbor in $S$. Denote the domination number, which is the size of any minimum dominating set of $G$, by $\gamma(G)$. For…

For a simple graph $G=(V,E)$ without any isolated vertex, a cosecure dominating set $D$ of $G$ satisfies the following two properties (i) $S$ is a dominating set of $G$, (ii) for every vertex $v \in S$ there exists a vertex $u \in V…

We consider (closed neighbourhood) packings and their generalization in graphs. A vertex set X in a graph G is a k-limited packing if for any vertex $v\in V(G)$, $\left|N[v] \cap X\right| \le k$, where N[v] is the closed neighbourhood of v.…

A subset $S \subseteq V$ in a graph $G = (V,E)$ is called a $[1, k]$-set, if for every vertex $v \in V \setminus S$, $1 \leq | N_G(v) \cap S | \leq k$. The $[1,k]$-domination number of $G$, denoted by $\gamma_{[1, k]}(G)$ is the size of 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…

Let $G$ be a simple graph with vertex set $V$ and edge set $E$, and let $S \subseteq V$. The \emph{open neighborhood} of $v \in V$, $N(v)$, is the set of vertices adjacent to $v$; the \emph{closed neighborhood} is given by $N[v] = N(v) \cup…

Let $G=\big{(}V(G),E(G)\big{)}$ be a graph with minimum degree $k$. A subset $S\subseteq V(G)$ is called a total $k$-dominating set if every vertex in $G$ has at least $k$ neighbors in $S$. Two disjoint sets $A,B\subset V(G)$ form a total…

A subset $S$ of vertices in a graph $G=(V, E)$ is a Dominating Set if each vertex in $V(G)\setminus S$ is adjacent to at least one vertex in $S$. Chellali et al. in 2013, by restricting the number of neighbors in $S$ of a vertex outside…

For a graph $G=(V,E)$, a subset $D$ of vertex set $V$, is a dominating set of $G$ if every vertex not in $D$ is adjacent to atleast one vertex of $D$. A dominating set $D$ of a graph $G$ with no isolated vertices is called a paired…

Given a simple, finite, nonempty graph $G=(V(G),E(G))$, a vertex subset $D\subseteq V(G)$ is said to be a dominating set if every vertex $v\in V(G)-D$ is adjacent to a vertex in $D$. The independent domination number $\gamma_i(G)$ is the…

Let $G = (V,E)$ be a simple, undirected and connected graph. A connected dominating set $S \subseteq V$ is a secure connected dominating set of $G$, if for each $ u \in V\setminus S$, there exists $v\in S$ such that $(u,v) \in E$ and the…

We consider (closed neighbourhood) packings and their generalization in graphs called limited packings. A vertex set X in a graph G is a k-limited packing if for any vertex $v\in V(G)$, $\left|N[v] \cap X\right| \le k$, where $N[v]$ is the…

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 set $D \subseteq V$ of a graph $G = (V,E)$ is called an outer-connected dominating set of $G$ if every vertex $v$ not in $D$ is adjacent to at least one vertex in $D$, and the induced subgraph of $G$ on $V \setminus D$ is connected. The…

Let $G=(V,E)$ be a graph. For a non-empty subset of vertices $S\subseteq V$, and vertex $v\in V$, let $\delta_S(v)=|\{u\in S:uv\in E\}|$ denote the cardinality of the set of neighbors of $v$ in $S$, and let $\bar{S}=V-S$. Consider the…

For a graph $G=(V,E)$, a set $D\subseteq V$ is called a \emph{disjunctive dominating set} of $G$ if for every vertex $v\in V\setminus D$, $v$ is either adjacent to a vertex of $D$ or has at least two vertices in $D$ at distance $2$ from it.…

A subset $D \subseteq V $of a graph $G = (V, E)$ is a $(1, j)$-set if every vertex $v \in V \setminus D$ is adjacent to at least $1$ but not more than $j$ vertices in D. The cardinality of a minimum $(1, j)$-set of $G$, denoted as…

Let $\mathcal{G}$ be a minor-closed graph class. We say that a graph $G$ is a $k$-apex of $\mathcal{G}$ if $G$ contains a set $S$ of at most $k$ vertices such that $G\setminus S$ belongs to $\mathcal{G}.$ We denote by $\mathcal{A}_k…

The study of power domination in graphs arises from the problem of placing a minimum number of measurement devices in an electrical network while monitoring the entire network. A power dominating set of a graph is a set of vertices from…

Given a graph $I=(V, E),$ $\emptyset \ne D \subseteq V,$ and an arbitrary nonempty set $X,$ an injective function $f: V\to 2^X \setminus \{\emptyset\}$ is an interference of $D$ with respect to $I,$ if for every vertex $u\in V\setminus D$…