Related papers: On global offensive alliance in zero-divisor graph…

Let $G=(V,E)$ be a simple graph. For a nonempty set $X\subset V,$ and a vertex $v\in V,$ $\delta_{X}(v)$ denotes the number of neighbors $v$ has in $X.$ A nonempty set $S\subset V$ is an \emph{offensive $r$-alliance} in $G$ if…

For a graph $G=(V,E)$, a set $S\subseteq V$ is a dominating set if every vertex in $V-S$ has at least a neighbor in $S$. A dominating set $S$ is a global offensive alliance if for each vertex $v$ in $V-S$ at least half the vertices from the…

An offensive alliance in a graph $\Gamma=(V,E)$ is a set of vertices $S\subset V$ where for every vertex $v$ in its boundary it holds that the majority of vertices in $v$'s closed neighborhood are in $S$. In the case of strong offensive…

A global offensive alliance in a graph $G$ is a set $S$ of vertices with the property that every vertex not belonging to $S$ has at least one more neighbor in $S$ than it has outside of $S$. The global offensive alliance number of $G$,…

An offensive alliance in a graph $\Gamma=(V,E)$ is a set of vertices $S\subset V$ where for every vertex $v$ in its boundary it holds that the majority of vertices in $v$'s closed neighborhood are in $S$. In the case of strong offensive…

Let $G=$ $\left( V,E\right) $ be a simple graph.\ A non-empty set $S \subseteq V$ is called a global offensive alliance if $S$ is a dominating set and for every vertex $v$ in $V-S$, at least half of the vertices from the closed neighborhood…

For a commutative ring $R$ with identity, the zero-divisor graph of $R$, denoted $\Gamma(R)$, is the graph whose vertices are the non-zero zero divisors of $R$ with two distinct vertices $x$ and $y$ are adjacent if and only if $xy=0$. In…

In this paper, we initiate the study of global offensive $k$-alliances in digraphs. Given a digraph $D=(V(D),A(D))$, a global offensive $k$-alliance in a digraph $D$ is a subset $S\subseteq V(D)$ such that every vertex outside of $S$ has at…

Let $\Gamma=(V,E)$ be a simple graph. For a nonempty set $X\subseteq V$, and a vertex $v\in V$, $\delta_{X}(v)$ denotes the number of neighbors $v$ has in $X$. A nonempty set $S\subseteq V$ is a \emph{defensive $k$-alliance} in…

The global defensive $k$-alliance is a very well studied notion in graph theory, it provides a method of classification of graphs based on relations between members of a particular set of vertices. In this paper we explore this notion in…

If $G=(V_G, E_G)$ is a graph, then $S\subseteq V_G$ is a global defensive $k$-alliance in $G$ if (i) each vertex not in $S$ has a neighbor in $S$ and (ii) each vertex of $S$ has at least $k$ more neighbors inside $S$ than outside of it. The…

Let $\Gamma=(V,E)$ be a simple graph. For a nonempty set $X\subseteq V$, and a vertex $v\in V$, $\delta_{X}(v)$ denotes the number of neighbors $v$ has in $X$. A nonempty set $S\subseteq V$ is a \emph{defensive $k$-alliance} in…

A defensive $k$-alliance in a graph is a set $S$ of vertices with the property that every vertex in $S$ has at least $k$ more neighbors in $S$ than it has outside of $S$. A defensive $k$-alliance $S$ is called global if it forms a…

A defensive alliance in an undirected graph $G=(V,E)$ is a non-empty set of vertices $S$ satisfying the condition that every vertex $v\in S$ has at least as many neighbours (including itself) in $S$ as it has in $V\setminus S$. We consider…

The Zero divisor Graph of a commutative ring $R$, denoted by $\Gamma[R]$, is a graph whose vertices are non-zero zero divisors of $R$ and two vertices are adjacent if their product is zero. In this paper we derive the Vertex and Edge…

A set $S\subseteq V$ of vertices is an offensive alliance in an undirected graph $G=(V,E)$ if each $v\in N(S)$ has at least as many neighbours in $S$ as it has neighbours (including itself) not in $S$. We study the classical and…

Let $G=(V,E)$ be a graph. A subset $S \subseteq V$ is called a global dominating set of $G$, if it serves as a dominating set in both $G$ and its complement $\overline{G}$. We define two disjoint subsets $V_1,V_2 \subseteq V$ to form a…

Let $G=(V,E)$ be a simple graph. A set $S\subseteq V$ is independent set of $G$, if no two vertices of $S$ are adjacent. The independence number $\alpha(G)$ is the size of a maximum independent set in the graph. %An independent set with…

In this paper, we prove an inequality on the cardinalities of the minimum size global defensive alliance and the minimum size global offensive alliance. A global defensive alliance is a dominating set such that when any point inside a…

A set $S$ of vertices of graph $G$ is a \textit{defensive alliance} of $G$ if for every $v \in S$, it holds $|N[v] \cap S| \geq |N[v]-S|$. An alliance $S$ is called $global$ if it is also a dominating set. In this paper, we determine the…