Ascending domination decomposition of subdivision of graphs

In this paper, the two major fields of graph theory namely decomposition and domination are connected and new concept called Ascending Domination Decomposition ($ADD$) of a graph $G$ is introduced. An $ADD$ of a graph $G$ is a collection $psi={G_{1},G_{2},ldots,G_{n}}$ of subgraphs of $G$ such that, each $G_{i}$ is connected, every edge of $G$ is in exactly one $G_{i}$ and $gamma(G_{i})=i$, $1leq ileq n$. In this paper, we prove the subdivision of some standard graphs admit $ADD$.