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 Ο={V1β,V2β,β¦,Vkβ} of
G is called a \emph{transitive k-partition} if Viβ dominates Vjβ for
all i,j, where 1β€i<jβ€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.Comment: arXiv admin note: text overlap with arXiv:2204.1314