Let $$G=(V, E)$$ be a graph where V and E are the vertex and edge sets, respectively. For two disjoint subsets A and B of V, we say A dominates B if every vertex of B is adjacent to at least one vertex of A in G. A vertex partition $$\pi = \{V_1, V_2, \ldots , V_k\}$$ of G is called a transitive partition of size k if $$V_i$$ dominates $$V_j$$ for all $$1\le i