Secure monophonic domination of graphs

Let G = (V, E) be a connected graph. A monophonic dominating set M is said to be a secure monophonic dominating set Sm (abbreviated as SMD set) of G if for each v∈V \M there exists u∈M such that v is adjacent to u and Sm = {M \(u)} ∪{v} is a monophonic dominating set. The minimum cardinality of a secure monophonic dominating set of G is the secure monophonic domination number of G and is denoted by γsm(G). In this paper, we investigate the secure monophonic domination number of subdivision of graphs such as subdivision of Path graph S(Pn), subdivision of Cycle graph S(Cn), subdivision of Star graph S(K1,n-1), subdivision Bistar graph S(Bm,n) and subdivision of Y-tree graph S(Yn+1).