Related papers: Quasiperfect domination in triangular lattices
A set $S$ of vertices in $G$ is a semitotal dominating set of $G$ if it is a dominating set of $G$ and every vertex in $S$ is within distance $2$ of another vertex of $S$. The \emph{semitotal domination number}, $\gamma_{t2}(G)$, is the…
Let $G$ be a graph of order $n$ and size $m$ and let $k\geq 1$ be an integer. A $k$-tuple total dominating set in $G$ is called a $k$-tuple total restrained dominating set of $G$ if each vertex $x\in V(G)-S$ is adjacent to at least $k$…
The open neighbourhood of a vertex $v$ of a graph $G$ is the set $N(v)$ consisting of all vertices adjacent to $v$ in $G$. For $D\subseteq V(G)$, we define $\overline{D}=V(G)\setminus D$. A set $D\subseteq V(G)$ is called a super dominating…
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 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 $\gamma(G)$…
In a graph $G$, a vertex dominates itself and its neighbours. A subset $S\subseteq V(G)$ is said to be a double dominating set of $G$ if $S$ dominates every vertex of $G$ at least twice. The minimum cardinality among all double dominating…
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…
Given a graph $G = (V,E)$, a \emph{perfect dominating set} is a subset of vertices $V' \subseteq V(G)$ such that each vertex $v \in V(G)\setminus V'$ is dominated by exactly one vertex $v' \in V'$. An \emph{efficient dominating set} is a…
A dominating set in a graph $G$ is a subset of vertices $D$ such that every vertex in $V\setminus D$ is a neighbor of some vertex of $D$. The domination number of $G$ is the minimum size of a dominating set of $G$ and it is denoted by…
A set $D$ 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 domination number of $G$ is the minimum cardinality of any total dominating set of $G$ and is denoted…
An open neighbourhood locating-dominating set is a set $S$ of vertices of a graph $G$ such that each vertex of $G$ has a neighbour in $S$, and for any two vertices $u,v$ of $G$, there is at least one vertex in $S$ that is a neighbour of…
A set $D \subseteq V(G)$ is a \emph{dominating set} of $G$ if every vertex not in $D$ is adjacent to at least one vertex in $D$. A dominating set of $G$ of minimum cardinality is called a $\gamma(G)$-set. For each vertex $v \in V(G)$, we…
A subset $D$ of the vertex set $V$ of a graph $G$ is called an $[1,k]$-dominating set if every vertex from $V-D$ is adjacent to at least one vertex and at most $k$ vertices of $D$. A $[1,k]$-dominating set with the minimum number of…
Let $\gamma_g(G)$ and $\gamma_{tg}(G)$ be the game domination number and the total game domination number of a graph $G$, respectively. Then $G$ is $\gamma_g$-perfect (resp. $\gamma_{tg}$-perfect), if every induced subgraph $F$ of $G$…
A set $S$ of vertices of a graph $G$ is \emph{distinguishing} if the sets of neighbors in $S$ for every pair of vertices not in $S$ are distinct. A \emph{locating-dominating set} of $G$ is a dominating distinguishing set. The…
Given a graph G = (V,E), a subset S of V is dominating if for every v in V - S there exists u in S such that uv is in E. A dominating subset S of V is secure if for every v in V - S there exists u in S such that (S - {u}) U {v} is…
Given a directed graph $D$, a set $S \subseteq V(D)$ is a total dominating set of $D$ if each vertex in $D$ has an in-neighbor in $S$. The total domination number of $D$, denoted $\gamma_t(D)$, is the minimum cardinality among all total…
The dominance complex $D(G)$ of a simple graph $G = (V,E)$ is the simplicial complex consisting of the subsets of $V$ whose complements are dominating. We show that the connectivity of $D(G)$ plus $2$ is a lower bound for the vertex cover…
Given a graph G=(V, E), a vertex is said to ve-dominate an edge if it is either incident with the edge or adjacent to one of its endpoints. A set of vertices is a ve-dominating set if it ve-dominates every edge of the graph. We introduce…
A $k$-tuple total dominating set ($k$TDS) of a graph $G$ is a set $S$ of vertices in which every vertex in $G$ is adjacent to at least $k$ vertices in $S$; the minimum size of a $k$TDS is denoted $\gamma_{\times k,t}(G)$. We give a…