On the monophonic rank of a graph

A set of vertices $S$ of a graph $G$ is $monophonically \ convex$ if every induced path joining two vertices of $S$ is contained in $S$. The $monophonic \ convex \ hull$ of $S$, $\langle S \rangle$, is the smallest monophonically convex set containing $S$. A set $S$ is $monophonic \ convexly \ independent$ if $v \not\in \langle S - \{v\} \rangle$ for every $v \in S$. The $monophonic \ rank$ of $G$ is the size of the largest monophonic convexly independent set of $G$. We present a characterization of the monophonic convexly independent sets. Using this result, we show how to determine the monophonic rank of graph classes like bipartite, cactus, triangle-free and line graphs in polynomial time. Furthermore, we show that this parameter can be computed in polynomial time for $1$-starlike graphs, $i.e.$, for split graphs, and that its determination is $NP$-complete for $k$-starlike graphs for any fixed $k \ge 2$, a subclass of chordal graphs. We also consider this problem on the graphs whose intersection graph of the maximal prime subgraphs is a tree.