Eccentric domination decomposition of graphs

Abstract

A decomposition $\left(G_1, G_2 \ldots . . G_n\right)$ of $G$ is said to be an eccentric domination decomposition (EDD) if i) $E(G)=E\left(G_1\right) \cup E\left(G_2\right) \cup \ldots \cup E\left(G_n\right)$ ii) Each $G_i$ is connected iii) $\gamma_{e d}\left(G_i\right)=i, i=1,2 \ldots n$. If a graph $G$ has EDD, we say that $G$ admits eccentric domination decomposition.