Complementary Nil Eccentric Domination Number of a Graph

A subset $D$ of the vertex set $V(G)$ of a graph $G$ is said to be a dominating set if every vertex not in $D$ is adjacent to atleast one vertex in $D$. A dominating set $D$ is said to be an eccentric dominating set if for every $v in V-D$, there exists atleast one eccentric point of $v$ in $D$. An eccentric dominating set $D$ of $G$ is a complementary nil eccentric dominating set if the induced subgraph $$ is not an eccentric dominating set for $G$. The minimum of the cardinalities of the complementary nil eccentric dominating sets of $G$ is called the complementary nil eccentric domination number $gamma_{cned}(G)$ of $G$. In this paper, bounds for $gamma_{cned}(G)$, its exact value for some particular classes of graphs and some results on complementary nil eccentric domination number are obtained.