Related papers: Domination and 2-degree-packing numbers in graphs

Let $G$ be a graph. A total dominating set of $G$ is a set $S$ of vertices of $G$ such that every vertex is adjacent to at least one vertex in $S$. The total domatic number of a graph is the maximum number of total dominating sets which…

In any graph $G$, the domination number $\gamma(G)$ is at most the independence number $\alpha(G)$. The Inverse Domination Conjecture says that, in any isolate-free $G$, there exists pair of vertex-disjoint dominating sets $D, D'$ with…

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…

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…

A set of vertices $X\subseteq V(G)$ is a $d$-distance dominating set if for every $u\in V(G)\setminus X$ there exists $x\in X$ such that $d(u,x) \le d$, and $X$ is a $p$-packing if $d(u,v) \ge p+1$ for every different $u,v\in X$. The…

Let $G$ be a graph. A dominating set $D\subseteq V(G)$ is a super dominating set if for every vertex $x\in V(G) \setminus D$ there exists $y\in D$ such that $N_G(y)\cap (V(G)\setminus D)) = \{x\}$. The cardinality of a smallest super…

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…

Let $G=(V,E)$ be a connected, finite undirected graph. A set $S \subseteq V$ is said to be a total dominating set of $G$ if every vertex in $V$ is adjacent to some vertex in $S$. The total domination number, $\gamma_{t}(G)$, is the minimum…

Let $G=(V,E)$ be a graph. A subset $D$ of $V(G)$ is called a super dominating set if for every $v \in V(G)-D$ there exists an external private neighbour of $v$ with respect to $V(G)-D.$ The minimum cardinality of a super dominating set is…

A set $S$ of vertices in a graph $G$ is a paired dominating set if every vertex of $G$ is adjacent to a vertex in $S$ and the subgraph induced by $S$ admits a perfect matching. The minimum cardinality of a paired dominating set of $G$ is…

A vertex subset $S$ of a graph $G=(V,E)$ is a $[1,2]$-dominating set if each vertex of $V\backslash S$ is adjacent to either one or two vertices in $S$. The minimum cardinality of a $[1,2]$-dominating set of $G$, denoted by…

The dominating number $\gamma(G)$ of a graph $G$ is the minimum size of a vertex set whose closed neighborhood covers all the vertices of the graph. The packing number $\rho(G)$ of $G$ is the maximum size of a vertex set whose closed…

A dominating set of a graph is a subset $D$ of its vertices such that every vertex not in $D$ is adjacent to at least one member of $D$. The domination number of a graph $G$ is the number of vertices in a smallest dominating set of $G$. The…

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…

A locating-dominating set of a graph $G$ is a dominating set $D$ of $G$ with the additional property that every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N(u)…

A subset $D$ of vertices of a graph $G$ is a \textit{dominating set} if for each $u\in V(G)\setminus D$, $u$ is adjacent to some vertex $v\in D$. The \textit{dominating number}, $\gamma(G)$ of $G$, is the minimum cardinality of a dominating…

A vertex subset $S$ of a graph $G$ is a double dominating set of $G$ if $|N[v]\cap S|\geq 2$ for each vertex $v$ of $G$, where $N[v]$ is the set of the vertex $v$ and vertices adjacent to $v$. The double domination number of $G$, denoted by…

A dominating set in a graph $G$ is a set $S$ of vertices of $G$ such that every vertex outside $S$ is adjacent to a vertex in $S$. A connected dominating set in $G$ is a dominating set $S$ such that the subgraph $G[S]$ induced by $S$ is…

Let $p$ be a positive integer and $G=(V,E)$ be a simple graph. A subset $D\subseteq V$ is a $p$-dominating set if each vertex not in $D$ has at least $p$ neighbors in $D$. The $p$-domination number $\g_p(G)$ is the minimum cardinality among…

For an integer $k \ge 1$, a (distance) $k$-dominating set of a connected graph $G$ is a set $S$ of vertices of $G$ such that every vertex of $V(G) \setminus S$ is at distance at most~$k$ from some vertex of $S$. The $k$-domination number,…