A set of vertices $S$ of a graph $G$ is {\em monophonically convex} if every induced path joining two vertices of $S$ is contained in $S$. The {\em monophonic convex hull of $S$}, $\langle S \rangle$, is the smallest monophonically convex set containing $S$. A set $S$ is {\em monophonic convexly independent} if $v \not\in \langle S - \{v\} \rangle$ for every $v \in S$. The {\em 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 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.

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关于图的单音秩

如果连接 $S$ 的两个顶点的每条诱导路径都包含在 $S$ 中，则图 $G$ 的一组顶点 $S$ 是 {\em 单音凸}。$S$} 的 {\em 单音凸包，$\langle S \rangle$，是包含 $S$ 的最小单音凸集。如果 $v \not\in \langle S - \{v\} \rangle$ 对于每个 $v \in S$，则集合 $S$ 是 {\em 单音凸独立}。$G$ 的 {\em monophonic rank} 是 $G$ 的最大单音凸独立集的大小。我们提出了单音凸独立集的特征。使用这个结果，我们展示了如何在多项式时间内确定二部图、仙人掌图、无三角形图和线图等图类的单音等级。此外，我们表明，对于 $1$-starlike 图，即对于分割图，可以在多项式时间内计算此参数，并且对于任何固定的 $k \ge 2$（弦图的一个子类），它的确定对于 $k$-starlike 图是 $\NP$-完全的。我们也在最大素数子图的交图是树的图上考虑这个问题。