Related papers: Constant-factor approximation of domination number…

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}…

Given a graph $G=(V,E)$, the dominating number of a graph is the minimum size of a vertex set, $V' \subseteq V$, so that every vertex in the graph is either in $V'$ or is adjacent to a vertex in $V'$. A Roman Dominating function of $G$ is…

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 graph $G$ is said to be $k$-$\gamma_{c}$-critical if the connected domination number $\gamma_{c}(G)$ is equal to $k$ and $\gamma_{c}(G + uv) < k$ for any pair of non-adjacent vertices $u$ and $v$ of $G$. Let $\zeta$ be the number of cut…

A dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex outside $D$ is adjacent to a vertex in $D$. A locating-dominating set of $G$ is a dominating set $D$ of $G$ with the additional property that every two…

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…

In this paper, we study the following problem: given a connected graph $G$, can we reduce the domination number of $G$ by at least one using $k$ edge contractions, for some fixed integer $k \geq 0$? We present positive and negative results…

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…

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 $\gamma(G)$…

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…

The $k$-cut problem asks, given a connected graph $G$ and a positive integer $k$, to find a minimum-weight set of edges whose removal splits $G$ into $k$ connected components. We give the first polynomial-time algorithm with approximation…

A subset $M$ of the edges of a graph $G$ is a matching if no two edges in $M$ are incident. A maximal matching is a matching that is not contained in a larger matching. A subset $S$ of vertices of a graph $G$ with no isolated vertices is 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…

The $k$-token graph of $G$ is the graph, $F_k(G)$, whose vertices are all the $k$-subsets of $V(G)$; with two of them adjacent whenever their symmetric difference is a pair of adjacent vertices in $G$. In this paper, we study the domination…

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…

We study the relationship between the eternal domination number of a graph and its clique covering number using both large-scale computation and analytic methods. In doing so, we answer two open questions of Klostermeyer and Mynhardt. We…

A vertex coloring of a graph $G$ is an assignment of colors to the vertices of $G$ such that every two adjacent vertices of $G$ have different colors. A coloring related property of a graphs is also an assignment of colors or labels to the…

Given a graph $G$, a dominating set $D$ is a set of vertices such that any vertex in $G$ has at least one neighbor (or possibly itself) in $D$. A ${k}$-dominating multiset $D_k$ is a multiset of vertices such that any vertex in $G$ has at…

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 a graph $G=(V,E)$ with no isolated vertices, a set $D\subseteq V$ is called a semipaired dominating set of G if $(i)$ $D$ is a dominating set of $G$, and $(ii)$ $D$ can be partitioned into two element subsets such that the vertices in…