Related papers: Complexity Issues Concerning the Quadruple Roman D…

An edge Roman dominating function of a graph $G$ is a function $f\colon E(G) \rightarrow \{0,1,2\}$ satisfying the condition that every edge $e$ with $f(e)=0$ is adjacent to some edge $e'$ with $f(e')=2$. The edge Roman domination number of…

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 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}…

For a graph $G$, let $\gamma_R(G)$ and $\gamma_{r2}(G)$ denote the Roman domination number of $G$ and the $2$-rainbow domination number of $G$, respectively. It is known that $\gamma_{r2}(G)\leq \gamma_R(G)\leq \frac{3}{2}\gamma_{r2}(G)$.…

We investigate the 2-domination number for grid graphs, that is the size of a smallest set $D$ of vertices of the grid such that each vertex of the grid belongs to $D$ or has at least two neighbours in $D$. We give a closed formula giving…

Given a graph $G=(V,E)$, a function $f:V\rightarrow \{0,1,2,3\}$ having the property that if $f(v)=0$, then there exist $ v_{1},v_{2}\in N(v)$ such that $f(v_{1})=f(v_{2})=2$ or there exists $ w \in N(v)$ such that $f(w)=3$, and if…

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 $G$.…

The concept of Roman domination has been a subject of intrigue for more than two decades with the fundamental Roman domination problem standing out as one of the most significant challenges in this field. This article studies a practically…

For a graph $G=(V,E)$ of order $n$, a Roman $\{2\}$-dominating function $f:V\rightarrow\{0,1,2\}$ has the property that for every vertex $v\in V$ with $f(v)=0$, either $v$ is adjacent to a vertex assigned $2$ under $f$, or $v$ is adjacent…

Let $G=(V,E)$ be a graph of order $n$ and let $\gamma _{R}(G)$ and $\partial (G)$ denote the Roman domination number and the differential of $G,$ respectively. In this paper we prove that for any integer $k\geq 0$, if $G$ is a graph of…

Let $G = (V,E)$ be a simple, undirected and connected graph. A connected (total) dominating set $S \subseteq V$ is a secure connected (total) dominating set of $G$, if for each $ u \in V \setminus S$, there exists $v \in S$ such that $uv…

The domination problem is a well-studied problem in graph theory. In this paper, we study two natural variants: the hop domination problem and the $2$-step domination problem. Let $G$ be a graph with vertex set $V$ and edge set $E$. For a…

Given a graph~$G$, the domination number, denoted by~$\gamma(G)$, is the minimum cardinality of a dominating set in~$G$. Dual to the notion of domination number is the packing number of a graph. A packing of~$G$ is a set of vertices whose…

Let $G=(V, E)$ be a simple and undirected graph. For some integer $k\geq 1$, a set $D\subseteq V$ is said to be a k-dominating set in $G$ if every vertex $v$ of $G$ outside $D$ has at least $k$ neighbors in $D$. Furthermore, for some real…

We study a variant of domination, called Roman domination, where we must assign to each vertex one of the labels 0, 1, or 2 and require that every vertex with label 0 has a neighbour with label 2. We study the problem of finding a low-cost…

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 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…

The signed double Roman domination problem is a combinatorial optimization problem on a graph asking to assign a label from $\{\pm{}1,2,3\}$ to each vertex feasibly, such that the total sum of assigned labels is minimized. Here feasibility…

Given a graph $G=(V(G), E(G))$, the size of a minimum dominating set, minimum paired dominating set, and a minimum total dominating set of a graph $G$ are denoted by $\gamma(G)$, $\gamma_{\rm pr}(G)$, and $\gamma_{t}(G)$, respectively. For…

For a graph $G=(V(G),E(G))$, an Italian dominating function (ID function) $f:V(G)\rightarrow\{0,1,2\}$ has the property that for every vertex $v\in V(G)$ with $f(v)=0$, either $v$ is adjacent to a vertex assigned $2$ under $f$ or $v$ is…