Abstract
For a connected graph of order at least two, the strong monophonic problem is to determine a smallest set of vertices of such that, by fixing one monophonic path between each pair of the vertices of all vertices of are covered. In this paper, certain general properties satisfied by the strong monophonic sets are studied. Also, the strong monophonic number of a several families of graphs and computational complexity are determined. AMS Subject Classification: 05C12, 05C82
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