Related papers: On Roman, Global and Restrained Domination in Grap…

Let $G=(V, E)$ be a simple undirected graph with no isolated vertex. A set $D_t\subseteq V$ is a total dominating set of $G$ if $(i)$ $D_t$ is a dominating set, and $(ii)$ the set $D_t$ induces a subgraph with no isolated vertex. The total…

In this paper we study the behaviour of the domination number of the Erd\H{o}s-R\'enyi random graph $\mathcal{G}(n,p)$. Extending a result of Wieland and Godbole we show that the domination number of $\mathcal{G}(n,p)$ is equal to one of…

The total domination number $\gamma_{t}(G)$ of a graph $G$ is the cardinality of a smallest set $D\subseteq V(G)$ such that each vertex of $G$ has a neighbor in $D$. The annihilation number $a(G)$ of $G$ is the largest integer $k$ such that…

Let $\gamma(G)$ and $i(G)$ be the domination number and the independent domination number of $G$, respectively. Rad and Volkmann posted a conjecture that $i(G)/ \gamma(G) \leq \Delta(G)/2$ for any graph $G$, where $\Delta(G)$ is its maximum…

The \emph{domination subdivision number} sd$(G)$ of a graph $G$ is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number of $G$. It has been shown…

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…

It is conjectured that the game domination number is at most $3n/5$ for every $n$-vertex graph which does not contain isolated vertices. It was proved in the recent years that the conjecture holds for several graph classes, including the…

Given a graph $G=(V,E)$, a function $f:V\to \{0,1,2\}$ is said to be a \emph{Roman Dominating function} if for every $v\in V$ with $f(v)=0$, there exists a vertex $u\in N(v)$ such that $f(u)=2$. A Roman Dominating function $f$ is said to be…

For a graph $G$, the $\gamma$-graph of $G$, $G(\gamma)$, is the graph whose vertices correspond to the minimum dominating sets of $G$, and where two vertices of $G(\gamma)$ are adjacent if and only if their corresponding dominating sets in…

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…

A total dominating set of a graph G with no isolated vertices is a subset S of the vertex set such that every vertex of G is adjacent to a vertex in S. The total domination number of G is the minimum cardinality of a total dominating set of…

A double Roman Dominating function on a graph $G$ is a function $ f:V\rightarrow \{0,1,2,3\}$ such that the following conditions hold. If $f(v)=0$, then vertex $v$ must have at least two neighbors in $V_2$ or one neighbor in $V_3$ and if…

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…

Given a graph $G=(V,E)$, a function $f:V\rightarrow \{0,1,2\}$ is a total Roman $\{2\}$-dominating function if: (1) every vertex $v\in V$ for which $f(v)=0$ satisfies that $\sum_{u\in N(v)}f(u)\geq 2$, where $N(v)$ represents the open…

The product power throttling number of a graph is defined to study product throttling for power domination. The domination number of a graph is an upper bound for its product power throttling number. It is established that the two…

The Roman dominating function on a graph $G=(V,E)$ is a function $f: V\rightarrow\{0,1,2\}$ such that each vertex $x$ with $f(x)=0$ is adjacent to at least one vertex $y$ with $f(y)=2$. The value $f(G)=\sum\limits_{u\in V(G)} f(u)$ is…

A domination coloring of a graph $G$ is a proper vertex coloring of $G$ such that each vertex of $G$ dominates at least one color class, and each color class is dominated by at least one vertex. The minimum number of colors among all…

For a graph $G = (V, E)$, a Roman dominating function $f : V \rightarrow \{0, 1, 2\}$ has the property that every vertex $v \in V $with $f (v) = 0$ has a neighbor $u$ with $f (u) = 2$. The weight of a Roman dominating function $f$ is the…

The well-known notion of domination in a graph abstracts the idea of protecting locations with guards. This paper introduces a new graph invariant, the autonomous domination number, which abstracts the idea of defending a collection of…

For a graph G, a signed domination function of G is a two-colouring of the vertices of G with colours +1 and -1 such that the closed neighbourhood of every vertex contains more +1's than -1's. This concept is closely related to…