On the monophonic rank of a graph

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