Related papers: Offensive alliances in cubic graphs

The 2-domination number $\gamma_2(G)$ of a graph $G$ is the minimum cardinality of a set $ D \subseteq V(G) $ for which every vertex outside $ D $ is adjacent to at least two vertices in $ D $. Clearly, $ \gamma_2(G) $ cannot be smaller…

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…

In this paper, we show that several graph parameters are known in different areas under completely different names. More specifically, our observations connect signed domination, monopolies, $\alpha$-domination, $\alpha$-independence,…

The power graph $\Gamma_G$ of a finite group $G$ is the graph with the vertex set $G$, where two distinct elements are adjacent if one is a power of the other. An $L(2, 1)$-labeling of a graph $\Gamma$ is an assignment of labels from…

Let $G(V,E)$ be a finite, simple, isolate-free graph. Two disjoint sets $A,B\subset V$ form a total coalition in $G$, if none of them is a total dominating set, but their union $A\cup B$ is a total dominating set. A vertex partition…

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $D\subseteq V$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is…

Let $G$ be a graph. A total dominating set of $G$ is a set $S$ of vertices of $G$ such that every vertex is adjacent to at least one vertex in $S$. The total domatic number of a graph is the maximum number of total dominating sets which…

A dominating set $S$ of a graph $G$ is called locating-dominating, LD-set for short, if every vertex $v$ not in $S$ is uniquely determined by the set of neighbors of $v$ belonging to $S$. Locating-dominating sets of minimum cardinality are…

A dominating set of a graph $G=(V,E)$ is a vertex set $D$ such that every vertex in $V(G) \setminus D$ is adjacent to a vertex in $D$. The cardinality of a smallest dominating set of $D$ is called the domination number of $G$ and is denoted…

A set $S$ of vertices of a graph is a defensive alliance if, for each element of $S$, the majority of its neighbours is in $S$. We consider the notion of local minimality in this paper. We are interested in locally minimal defensive…

The Fibonacci cube $\Gamma_n$ is is the graph whose vertices are independent subsets of the path graph of length $n$, where two such vertices are considered adjacent if they differ by the addition or removal of a single element. Klav\v{z}ar…

A subset $D$ of vertices of a graph $G$ is a total dominating set if every vertex of $G$ is adjacent to at least one vertex of $D$. The total dominating set $D$ is called a total co-independent dominating set if the subgraph induced by…

Let $A$ be a group acting by automorphisms on the group $G.$ \textit{The commuting graph $\Gamma(G,A)$ of $A$-orbits} of this action is the simple graph with vertex set $\{x^{A} : 1\ne x \in G \}$, the set of all $A$-orbits on $G\setminus…

Given a graph $G=(V,E)$, $S\subseteq V$ is a dominating set if every $v\in V\setminus S$ is adjacent to an element of $S$. The Minimum Dominating Set problem asks for a dominating set with minimum cardinality. It is well known that its…

If $\Gamma$ is a graph for which every edge is in exactly one clique of order $\omega$, then one can form a new graph with vertex set equal to these cliques. This is a generalization of the line graph of $\Gamma$. We discover many general…

A subset $D$ of $V$ is \emph{dominating} in $G$ if every vertex of $V-D$ has at least one neighbour in $D;$ let $\gamma(G)$ be the minimum cardinality among all dominating sets in $G.$ A graph $G$ is $\gamma$-$q$-{\it critical} if the…

In any graph $G$, the domination number $\gamma(G)$ is at most the independence number $\alpha(G)$. The Inverse Domination Conjecture says that, in any isolate-free $G$, there exists pair of vertex-disjoint dominating sets $D, D'$ with…

The dominating number $\gamma(G)$ of a graph $G$ is the minimum size of a vertex set whose closed neighborhood covers all the vertices of the graph. The packing number $\rho(G)$ of $G$ is the maximum size of a vertex set whose closed…

A set $D$ of vertices is a strong dominating set in a graph $G$, if for every vertex $x\in V(G) \setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $deg(x) \leq deg(y)$. The strong domination number $\gamma_{st}(G)$ of $G$ is the…

The intersection ideal graph $\Gamma(S)$ of a semigroup $S$ is a simple undirected graph whose vertices are all nontrivial left ideals of $S$ and two distinct left ideals $I, J$ are adjacent if and only if their intersection is nontrivial.…