On Near Equitable Domination in Graphs
ali mohammed sahl · Asian Journal of Current Engineering and Maths · 2014
Let $G=(V,E)$ be a graph, $D\subseteq V(G)$ and $u$ be any vertex in $D$. The out degree of $u$ with respect to $D$ denoted by $ od_{_{D}}(u)$, is defined as $od_{_{D}}(u)=|N(u)\D)|$. Let $G=(V,E)$ be a graph, $D\subseteq V(G)$ and $u$ be any vertex in $D$. The out degree of $u$ with respect to $D$ denoted by $ od_{_{D}}(u)$, is defined as $od_{_{D}}(u)=|N(u)\cap (V-D)|$. $D$ is called a near equitable dominating set of $G$ if for every $v\in V-D$, there exists a vertex $u\in D$ such that $u$ is adjacent to $v$ and $|od_{_D}(u)-od_{_{V-D}}(v)|\leq 1$. The minimum cardinality of such a dominating set is denoted by $\gamma_{ne}$ and is called the near equitable domination number of $G$. In this paper, we introduce the concept of near equitable domination. The minimal near equitable dominating sets are established. The relation between $\gamma_{ne}(G)$, $\gamma_{e}(G)$ and $\gamma(G)$ are obtained, bounds for $\gamma_{ne}(G)$ are found. Near equitable domatic partition in a graph $G$ is also studied.