Strong Total Monophonic Problems in Product Graphs, Networks, and Its Computational Complexity

Let G be a graph with vertex set as VG and edge set as EG which is simple as well as connected. The problem of strong total monophonic set is to find the set of vertices T⊆VG, which contains no isolated vertices, and all the vertices in VG\T lie on a fixed unique chordless path between the pair of vertices in T. The cardinality of strong total monophonic set which is minimum is defined as strong total monophonic number, denoted by smtG. We proved the NP-completeness of strong total monophonic set for general graphs. The strong total monophonic number of certain graphs and networks is derived. If l,m,n are positive integers with 5≤l≤m≤n and m≤2l−1, then we can construct a connected graph G with strong monophonic number l and strong total monophonic number m.