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
For a graph G(V (G), E(G)), the set Z ⊆ V (G) is called a k-geodetic propagation set if Z is both a k-geodetic set as well as a propagating set. The cardinality of the minimum k-geodetic propagation set is called the k-geodetic propagation number. Few general results and also the computational complexity part of this concept for general,bipartite and chordal graphs are derived.
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