Abstract
Let G be a graph with vertex set as V(G) and edge set as E(G) which is simple as well as connected. The problem of strong total monophonic set is to find the set of vertices T⊆V(G), which contains no isolated vertices, and all the vertices in V(G)\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 smt(G). 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.
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