Related papers: On $(t,r)$ broadcast domination of certain grid gr…

A vertex subset $S$ in a graph $G$ is a dominating set if every vertex not contained in $S$ has a neighbor in $S$. A dominating set $S$ is a connected dominating set if the subgraph $G[S]$ induced by $S$ is connected. A connected dominating…

A set $S$ of vertices in a graph $G$ is a total dominating set of $G$ if every vertex is adjacent to a vertex in $S$. The total domination number $\gamma_t(G)$ is the minimum cardinality of a total dominating set of $G$. The total…

An independent broadcast on a graph $G$ is a function $f: V \longrightarrow \{0,\ldots,{\rm diam}(G)\}$ such that $(i)$ $f(v)\leq e(v)$ for every vertex $v\in V(G)$, where $\operatorname{diam}(G)$ denotes the diameter of $G$ and $e(v)$ the…

The independent domination number $i(G)$ of a graph $G$ is the minimum cardinality of a maximal independent set of $G$, also called an $i(G)$-set. The $i$-graph of $G$, denoted $\mathcal{I}(G)$, is the graph whose vertices correspond to the…

For some $\alpha$ with $0 < \alpha \le 1$, a subset $X$ of vertices in a graph $G$ of order~$n$ is an $\alpha$-partial dominating set of $G$ if the set $X$ dominates at least $\alpha \times n$ vertices in $G$. The $\alpha$-partial…

A set $W\subseteq V(G)$ is called a resolving set, if for each two distinct vertices $u,v\in V(G)$ there exists $w\in W$ such that $d(u,w)\neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The minimum cardinality…

A mixed dominating set of a graph $G = (V, E)$ is a mixed set $D$ of vertices and edges, such that for every edge or vertex, if it is not in $D$, then it is adjacent or incident to at least one vertex or edge in $D$. The mixed domination…

A set $D$ of vertices of a graph $G$ is a dominating set if each vertex of $V(G)\setminus D$ is adjacent to some vertex of $D$. The domination number of $G$, $\gamma(G)$, is the minimum cardinality of a dominating set of $G$. A graph $G$ is…

A subset $S \subseteq V$ in a graph $G = (V,E)$ is called a $[1, k]$-set, if for every vertex $v \in V \setminus S$, $1 \leq | N_G(v) \cap S | \leq k$. The $[1,k]$-domination number of $G$, denoted by $\gamma_{[1, k]}(G)$ is the size of the…

For a graph $G$, the central graph $C(G)$ is the graph constructed from $G$ by subdividing each edge of $G$ with one vertex and also by adding an edge to every pair of non-adjacent vertices in $G$. Also for a graph $G$, let $\gamma(G)$ and…

Let $G = (V, E)$ be a graph. A set $S \subseteq V$ is a restrained dominating set (RDS) if every vertex not in $S$ is adjacent to a vertex in $S$ and to a vertex in $V - S$. The restrained domination number of $G$, denoted by $\gamma_r(G)$,…

Let $G=(V,E)$ be a simple graph. A set $D\subseteq V$ is a strong dominating set of $G$, if for every vertex $x\in V\setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $deg(x)\leq deg(y)$. The strong domination number…

The neighbourhood of a vertex $v$ of a graph $G$ is the set $N(v)$ 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 set if for every…

Let $G$ be a connected graph. A non-empty $S\subseteq V(G)$ is a $2$-movable dominating set of $G$ if $S$ is a dominating set and for every pair $x,y \in S$, $S\backslash \{x, y\}$ is a dominating set in $G$, or there exist $u, v \in V(G)…

In this paper we initiate the study of broadcast dimension, a variant of metric dimension. Let $G$ be a graph with vertex set $V(G)$, and let $d(u,w)$ denote the length of a $u-w$ geodesic in $G$. For $k \ge 1$, let $d_k(x,y)=\min \{d(x,y),…

Given a graph $G$, a subset $S$ of vertices of $G$ is an efficient dominating set ($EDS$) if $|N[v] \cap S|=1,$ for all $v\in V(G)$. A graph $G$ is efficiently dominatable if it possesses an $EDS$. The efficient domination number of G is…

The Grundy domination number of a graph $G = (V,E)$ is the length of the longest sequence of unique vertices $S = (v_1, \ldots, v_k)$ satisfying $N[v_i] \setminus \cup_{j=1}^{i-1}N[v_j] \neq \emptyset$ for each $i \in [k]$. Recently, a…

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…

In the Telephone Broadcast problem we are given a graph $G=(V,E)$ with a designated source vertex $s\in V$. Our goal is to transmit a message, which is initially known only to $s$, to all vertices of the graph by using a process where in…

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…