Related papers: Complexity results for $k$-domination and $\alpha$…
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 $G=(V,E)$ 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 vertex set $V$, and let $k$ be a positive integer. A set $D \subseteq V$ is a \emph{distance-$k$ dominating set} of $G$ if, for each vertex $u \in V-D$, there exists a vertex $w\in D$ such that $d(u,w) \le k$, where…
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…
In a graph $G$, a vertex dominates itself and its neighbours. A set $D\subseteq V(G)$ is said to be a $k$-tuple dominating set of $G$ if $D$ dominates every vertex of $G$ at least $k$ times. The minimum cardinality among all $k$-tuple…
Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is…
For $k \geq 1$, in a graph $G=(V,E)$, a set of vertices $D$ is a distance $k$-dominating set of $G$, if any vertex in $V\setminus D$ is at distance at most $k$ from some vertex in $D$. The minimum cardinality of a distance $k$-dominating…
For $k \geq 1$ and a graph $G$ without isolated vertices, a \emph{total (distance) $k$-dominating set} of $G$ is a set of vertices $S \subseteq V(G)$ such that every vertex in $G$ is within distance $k$ to some vertex of $S$ other than…
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…
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 $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is…
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 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,…
A dominating set of a graph $G=(V,E)$ is a vertex set $D$ such that every vertex in $V(G) \setminus D$ is adjacent to a vertex in $D$. The cardinality of a smallest dominating set of $D$ is called the domination number of $G$ and is denoted…
For $k \ge 1$ an integer, a set $S$ of vertices in a graph $G$ with minimum degree at least~$k-1$ is a $k$-tuple dominating set of $G$ if every vertex of $S$ is adjacent to at least $k-1$ vertices in $S$ and every vertex of $V(G) \setminus…
Let $k$ be a positive integer and $G=(V,E)$ be a graph of minimum degree at least $k-1$. A function $f:V\rightarrow \{-1,1\}$ is called a \emph{signed $k$-dominating function} of $G$ if $\sum_{u\in N_G[v]}f(u)\geq k$ for all $v\in V$. The…
For a graph G=(V,E), the k-dominating graph of G, denoted by $D_{k}(G)$, has vertices corresponding to the dominating sets of G having cardinality at most k, where two vertices of $D_{k}(G)$ are adjacent if and only if the dominating set…
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…
Given a positive integer $k$, a $k$-dominating set in a graph $G$ is a set of vertices such that every vertex not in the set has at least $k$ neighbors in the set. A total $k$-dominating set, also known as a $k$-tuple total dominating set,…
A set $S$ of vertices in a graph $G$ is a dominating set if every vertex of $G$ is in $S$ or is adjacent to a vertex in $S$. If, in addition, $S$ is an independent set, then $S$ is an independent dominating set. The domination number…