Secure monophonic domination number of product of graphs

Abstract

Let G be a graph which is connected. A monophonic dominating set M is said to be a secure monophonic dominating set Sm(written as SMD set) of G if for each g ∈ V \M there exists f ∈ M such that g is adjacent to f and Sm = (M\{f} ∪ {g} is a monophonic dominating set. The smallest cardinality of a secure monophonic dominating set of G is the secure monophonic domination number of G and the notation is γsm(G). In this article, we discuss the secure monophonic domination number of product of graphs.