Related papers: An Efficient Algorithm for Mixed Domination on Gen…

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…

Say that an edge of a graph $G$ dominates itself and every other edge adjacent to it. An edge dominating set of a graph $G=(V,E)$ is a subset of edges $E' \subseteq E$ which dominates all edges of $G$. In particular, if every edge of $G$ is…

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…

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

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…

A set $D \subseteq V$ for the graph $G=(V, E)$ is called a dominating set if any vertex $v\in V\setminus D$ has at least one neighbor in $D$. Fomin et al.[9] gave an algorithm for enumerating all minimal dominating sets with $n$ vertices in…

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 $G(V,E)$ be a simple, undirected and connected graph. A dominating set $S \subseteq V(G)$ is called a $2$-\textit{secure dominating set} ($2$-SDS) in $G$, if for every pair of distinct vertices $u_1,u_2 \in V(G)$ there exists a pair of…

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.…

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…

Counting dominating sets in a graph $G$ is closely related to the neighborhood complex of $G$. We exploit this relation to prove that the number of dominating sets $d(G)$ of a graph is determined by the number of complete bipartite…

Given a graph $G = (V, E)$ and an integer $k$, the Minimum Membership Dominating Set problem asks to compute a set $S \subseteq V$ such that for each $v \in V$, $1 \leq |N[v] \cap S| \leq k$. The problem is known to be NP-complete even on…

Given a graph $G=(V,E)$, $S\subseteq V$ is a dominating set if every $v\in V\setminus S$ is adjacent to an element of $S$. The Minimum Dominating Set problem asks for a dominating set with minimum cardinality. It is well known that its…

The domination polynomial D(G,x) is the ordinary generating function for the dominating sets of an undirected graph G=(V,E) with respect to their cardinality. We consider in this paper representations of D(G,x) as a sum over subsets of the…

The domination problem and its variants represent a classical domain within algorithmic graph theory. Among these variants, the paired-domination problem holds particular prominence due to its real-world implications in security and…

A subset $S$ of vertices of $G$ is a \textit{dominating set} of $G$ if every vertex in $V(G)-S$ has a neighbor in $S$. The \textit{domination number} \(\gamma(G)\) is the minimum cardinality of a dominating set of $G$. A dominating set $S$…

A dominating set $D$ for a graph $G$ is a subset of $V(G)$ such that any vertex not in $D$ has at least one neighbor in $D$. The domination number $\gamma(G)$ is the size of a minimum dominating set in $G$. Vizing's conjecture from 1968…

A dominating set of a graph $G$ is a set $D\subseteq V(G)$ such that \-every vertex of $G$ is either in $D$ or is adjacent to a vertex in $D$. The domination number of $G$, $\gamma(G)$, is the minimum order of a dominating set. A subset $R$…

A subset $S\subseteq V$ in a graph $G=(V,E)$ is a $k$-quasiperfect dominating set (for $k\geq 1$) if every vertex not in $S$ is adjacent to at least one and at most $k$ vertices in $S$. The cardinality of a minimum $k$-quasiperfect…

Let $G=(V,E)$ be a finite undirected graph. An edge set $E' \subseteq E$ is a {\em dominating induced matching} ({\em d.i.m.}) in $G$ if every edge in $E$ is intersected by exactly one edge of $E'$. The \emph{Dominating Induced Matching}…