$[1, 2]$- AND $(1, 2)^\ast$-DOMINATING SETS IN SOME SPECIAL GRAPHS

Let $G=(V, E)$ be a simple graph. Then a subset $S\subseteq V(G)$ is a $[1, 2]$-set if for each $v \in V\setminus S$, $1 \le |N(v) \cap S| \le 2$. A minimum cardinality $\gamma_{[1, 2]}(G)$ of a $[1, 2]$-set is called a $[1, 2]$-domination number of $G$. Also, $S$ is called a $(1, 2)^\ast$-dominating set of $G$ if $S$ is a dominating set and for every $v \in V\setminus S$, there exists $u \in S$ such that $d_G(u, v)=2$. A minimum cardinality $\gamma^\ast_{(1, 2)}(G)$ of such a set is called a $(1, 2)^\ast$-domination number of $G$. In this paper, we characterize the $[1, 2]$-dominating sets and $(1, 2)^\ast$-dominating sets in some special graphs such as Cocktail Party Graph, Crown Graph, and Gear Graph. Correspondingly, the $[1, 2]$-domination number and $(1, 2)^\ast$-domination number are also obtained.