monophonic domination integrity in fuzzy graphs

Let m be a subset of v (g) and let g : (v, σ, μ) be a fuzzy graph. the monophonic domination integrity (mdi) of g is defined by (mdi) ̃(g)^= min{|m|+m(g−m): m is a monophonic dominating set of g}, where |m|=∑_(u∈m)σ(u)and m(g − m) is the order of the greatest component of g−m. the notion of vulnerability parameter mdi in fuzzy graphs is presented in this work. further, the mdi for complete fuzzy graph, complete bipartite fuzzy graph, join and cartesian product of two fuzzy graphs and bounds are also discussed. also, we present a decision-making problem involving the optimization of bus routes and the strategic placement of bus stations using mdi principles.