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

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 s m t 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 ≤ 2 l − 1 , then we can construct a connected graph G with strong monophonic number l and strong total monophonic number m .