  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 $A$
\textit{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 \emph{transitive $k$-partition} if $V_i$ dominates $V_j$ for
all $i,j$, where $1\leq i<j\leq k$. The maximum integer $k$ for which the above
partition exists is called \emph{transitivity} of $G$ and it is denoted by
$Tr(G)$. The \textsc{Maximum Transitivity Problem} is to find a transitive
partition of a given graph with the maximum number of partitions. It was known
that the decision version of \textsc{Maximum Transitivity Problem} is
NP-complete for chordal graphs [Iterated colorings of graphs, \emph{Discrete
Mathematics}, 278, 2004]. In this paper, we first prove that this problem can
be solved in linear time for \emph{split graphs} and for the \emph{complement
of bipartite chain graphs}, two subclasses of chordal graphs. We also discuss
Nordhaus-Gaddum type relations for transitivity and provide counterexamples for
an open problem posed by J. T. Hedetniemi and S. T. Hedetniemi [The
transitivity of a graph, \emph{J. Combin. Math. Combin. Comput}, 104, 2018].
Finally, we characterize transitively critical graphs having fixed
transitivity.
