Related papers: On the complexity of Dominating Set for graphs wit…

Let G be a finite undirected graph. A vertex dominates itself and all its neighbors in G. A vertex set D is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient…

For a graph $G=(V,E)$, a set $D \subseteq V$ is called a semitotal dominating set of $G$ if $D$ is a dominating set of $G$, and every vertex in $D$ is within distance~$2$ of another vertex of~$D$. The \textsc{Minimum Semitotal Domination}…

Let $G=(V,E)$ be a graph. A subset $D\subseteq V$ is a dominating set if every vertex not in $D$ is adjacent to a vertex in $D$. The domination number of $G$, denoted by $\gamma(G)$, is the smallest cardinality of a dominating set of $G$.…

A vertex set $D$ in a finite undirected graph $G$ is an efficient dominating set (e.d.s. for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The Efficient Domination (ED) problem, which asks for the existence…

Let $G=(V,E)$ be a graph. Let $w$ be a positive integer. A $w$-dominating set is a vertex subset $S$ such that for all $v\in V$, either $v\in S$ or it has at least $w$ neighbors in $S$. The $w$-Dominating Set problem is to find the minimum…

A dominating set of a graph $G$ is a set $S \subseteq V(G)$ such that every vertex in $V(G) \setminus S$ has a neighbor in $S$, where two vertices are neighbors if they are adjacent. A secure dominating set of $G$ is a dominating set $S$ of…

A mixed dominating set $S$ of a graph $G=(V,E)$ is a subset $ S \subseteq V \cup E$ such that each element $v\in (V \cup E) \setminus S$ is adjacent or incident to at least one element in $S$. The mixed domination number $\gamma_m(G)$ of a…

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 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…

In the Independent set problem, the input is a graph $G$, every vertex has a non-negative integer weight, and the task is to find a set $S$ of pairwise non-adjacent vertices, maximizing the total weight of the vertices in $S$. We give an…

A set $S$ of vertices in a graph $G(V,E)$ is called a dominating set if every vertex $v\in V$ is either an element of $S$ or is adjacent to an element of $S$. A set $S$ of vertices in a graph $G(V,E)$ is called a total dominating set if…

Given a graph $G$, a dominating set of $G$ is a set $S$ of vertices such that each vertex not in $S$ has a neighbor in $S$. The domination number of $G$, denoted $\gamma(G)$, is the minimum size of a dominating set of $G$. The independent…

A set $D$ of vertices in an isolate-free graph $G$ is a semitotal dominating set of $G$ if $D$ is a dominating set of $G$ and every vertex in $D$ is within distance $2$ from another vertex of $D$.The semitotal domination number of $G$ is…

A vertex set $D$ in a finite undirected graph $G$ is an {\em efficient dominating set} (e.d.s.\ for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \emph{Efficient Domination} (ED) problem, which asks for…

In a graph $G$, an efficient dominating set is a subset $D$ of vertices such that $D$ is an independent set and each vertex outside $D$ has exactly one neighbor in $D$. The Minimum Weight Efficient Dominating Set (Min-WED) problem asks for…

Given a graph $G=(V,E)$ and an integer $k$, the Minimum Membership Dominating Set (MMDS) problem seeks to find a dominating set $S \subseteq V$ of $G$ such that for each $v \in V$, $|N[v] \cap S|$ is at most $k$. We investigate the…

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 $G$ be a finite undirected graph. A vertex {\em dominates} itself and all its neighbors in $G$. A vertex set $D$ is an {\em efficient dominating set} (\emph{e.d.}\ for short) of $G$ if every vertex of $G$ is dominated by exactly one…

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…

In a finite undirected graph $G=(V,E)$, a vertex $v \in V$ {\em dominates} itself and its neighbors. A vertex set $D \subseteq V$ in $G$ is an {\em efficient dominating set} ({\em e.d.} for short) of $G$ if every vertex of $G$ is dominated…