  A vertex partition $\pi = \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a
\emph{transitive partition} of size $k$ if $V_i$ dominates $V_j$ for all $1\leq
i<j\leq k$. For two disjoint subsets $A$ and $B$ of $V$, we say $A$
\emph{strongly dominates} $B$ if for every vertex $y\in B$, there exists a
vertex $x\in A$, such that $xy\in E$ and $deg_G(x)\geq deg_G(y)$. A vertex
partition $\pi = \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{strong
transitive partition} of size $k$ if $V_i$ strongly dominates $V_j$ for all
$1\leq i<j\leq k$. The \textsc{Maximum Strong Transitivity Problem} is to find
a strong transitive partition of a given graph with the maximum number of
parts. In this article, we initiate the study of this variation of transitive
partition from algorithmic point of view. We show that the decision version of
this problem is NP-complete for chordal graphs. On the positive side, we prove
that this problem can be solved in linear time for trees and split graphs.
