On Near Equitable Domination in Graphs

Abstract

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.