Tournament transitivity of graphs

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 partition} of size $k$ if $V_i$ dominates $V_j$ for all $1\leq i<j\leq k$. A vertex partition $\pi = \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{tournament transitive partition} of size $k$ if $V_i$ dominates $V_j$ for all $1\leq i<j\leq k$ and $V_j$ does not dominate $V_i$ for $i<j$. The maximum integer $k$ for which the above partition exists is called \emph{tournament transitivity} of $G$, and it is denoted by $TTr(G)$. The \textsc{Maximum Tournament Transitivity Problem} is to find a tournament transitive partition of a given graph with the maximum number of parts. In this article, we study this variation of transitive partition from a structure and algorithmic point of view. We show that the decision version of this problem is NP-complete for chordal graphs (connected), perfect elimination bipartite graphs (disconnected) and doubly chordal graphs (disconnected). On the positive side, we prove that this problem can be solved in polynomial time for trees. Furthermore, we characterize \textup{Type-I BCG} with equal transitivity and tournament transitivity and find some sufficient conditions under which the above two parameters are equal for a \textup{Type-II BCG}. Finally, we show that for \textup{Type-III BCG}, these two parameters are never equal.