  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 $A$
\emph{dominates} $B$ if every vertex of $B$ is adjacent to at least one vertex
of $A$. 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 $1\leq
i<j\leq k$. In this article, we initiate the study of a generalization of
transitive partition, namely \emph{$d_2$-transitive partition}. For two
disjoint subsets $A$ and $B$ of $V$, we say $A$ \emph{$d_2$-dominates} $B$ if,
for every vertex of $B$, there exists a vertex in $A$, such that the distance
between them is at most two. A vertex partition $\pi = \{V_1, V_2, \ldots,
V_k\}$ of $G$ is called a \emph{$d_2$-transitive partition} of size $k$ if
$V_i$ $d_2$-dominates $V_j$ for all $1\leq i<j\leq k$. The maximum integer $k$
for which the above partition exists is called \emph{$d_2$-transitivity} of
$G$, and it is denoted by $Tr_{d_2}(G)$. The \textsc{Maximum $d_2$-Transitivity
Problem} is to find a $d_2$-transitive partition of a given graph with the
maximum number of parts. We show that this problem can be solved in linear time
for the complement of bipartite graphs and bipartite chain graphs. On the
negative side, we prove that the decision version of the \textsc{Maximum
$d_2$-Transitivity Problem} is NP-complete for split graphs, bipartite graphs,
and star-convex bipartite graphs.
