On the monophonic rank of a graph
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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, ⟨ S ⟩, is the smallest monophonically convex set containing S. A set S is monophonic convexly independent if v ∉⟨ S - {v}⟩ for every v ∈ 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 computed in polynomial time for 1-starlike graphs, i.e., for split graphs, and that its determination is -complete for k-starlike graphs for any fixed k ≥ 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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