Related papers: Fair coalition in graphs
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…
The coalition in a graph $G$ consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a dominating set but whose union $V_{1}\cup V_{2}$, is a dominating set. A coalition partition in a graph $G$ is a vertex…
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…
In this paper, we propose and investigate the concept of $k$-coalitions in graphs, where $k\ge 1$ is an integer. A $k$-coalition refers to a pair of disjoint vertex sets that jointly constitute a $k$-dominating set of the graph, meaning…
A secure coalition in a graph $G$ consists of two disjoint vertex sets $V_1$ and $V_2$, neither of which is a secure dominating set, but whose union $V_1 \cup V_2$ forms a secure dominating set. A secure coalition partition…
A set $D$ of vertices of a graph $G$ is a dominating set of $G$ if every vertex in $V_G-D$ is adjacent to at least one vertex in $D$. The domination number (upper domination number, respectively) of a graph $G$, denoted by $\gamma(G)$…
A set $S$ of vertices in a graph $G(V,E)$ is called a dominating set if every vertex $v\in V$ is either an element of $S$ or is adjacent to an element of $S$. A set $S$ of vertices in a graph $G(V,E)$ is called a total dominating set if…
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…
A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex that does not belong to $S$ is adjacent to a vertex in $S$. The domination number $\gamma(G)$ of $G$ is the minimum cardinality of a dominating set of $G$. The…
Let $G=(V,E)$ be a graph. A set $S\subseteq V(G)$ is a dominating set, if every vertex in $V(G)\backslash S$ is adjacent to at least one vertex in $S$. The $k$-dominating graph of $G$, $D_k (G)$, is defined to be the graph whose vertices…
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 dominating set of a graph $G$ is a set $D\subseteq V_G$ such that every vertex in $V_G-D$ is adjacent to at least one vertex in $D$, and the domination number $\gamma(G)$ of $G$ is the minimum cardinality of a dominating set of $G$. In…
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…
A set $D$ of vertices in a graph $G = (V, E)$ is a locating-dominating set (LD-set) if it is dominating and every two vertices $u$, $v$ of $V\setminus D$ satisfy $N(u) \cap D \neq N(v) \cap D$. Two disjoint sets $A,B\subset V(G)$ form a…
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…
Let $G$ be graph with vertex set $V(G)$ and order $n$. A coalition in a graph $G$ consists of two disjoint sets of vertices $V_1$ and $V_2$, neither of which is a dominating set but whose union $V_1 \cup V_2$ is a dominating set. A…
Given a graph $G$, the $k$-dominating graph of $G$, $D_k(G)$, is defined to be the graph whose vertices correspond to the dominating sets of $G$ that have cardinality at most $k$. Two vertices in $D_k(G)$ are adjacent if and only if the…
A graph is subcubic if it is connected and its maximum vertex degree does not exceed 3. Two disjoint vertex subsets of a graph $G$ form a connected coalition in $G$ if neither of them is a connected dominating set but their union is a…
A set $D$ of vertices in an isolate-free graph $G$ is a semitotal dominating set of $G$ if $D$ is a dominating set of $G$ and every vertex in $D$ is within distance $2$ from another vertex of $D$.The semitotal domination number of $G$ is…