Algorithmic study on $2$-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 $A$ \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 study a variation of transitive partition, namely \emph{$2$-transitive partition}. For two disjoint subsets $A$ and $B$ of $V$, we say $A$ \emph{$2$-dominates} $B$ if every vertex of $B$ is adjacent to at least two vertices of $A$. A vertex partition $\pi = \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{$2$-transitive partition} of size $k$ if $V_i$ $2$-dominates $V_j$ for all $1\leq i<j\leq k$. The \textsc{Maximum $2$-Transitivity Problem} is to find a $2$-transitive partition of a given graph with the maximum number of parts. We show that the decision version of this problem is NP-complete for chordal and bipartite graphs. On the positive side, we design three linear-time algorithms for solving \textsc{Maximum $2$-Transitivity Problem} in trees, split and bipartite chain graphs.