Perfect double Italian domination of a graph

Abstract

For a graph G = ( V , E ) with V = V ( G ) and E = E ( G ) , a perfect double Italian dominating function is a function f : V → { 0 , 1 , 2 , 3 } having the property that 3 ≤ ∑ u ∈ N G [ v ] f ( u ) ≤ 4 , for every vertex v ∈ G with f ( v ) ∈ { 0 , 1 } . The weight of a perfect double Italian dominating function f is the sum f ( V ) = ∑ v ∈ V ( G ) f ( v ) and the minimum weight of a perfect double Italian dominating function on G is the perfect double Italian domination number γ d I p ( G ) of G. We initiate the study of perfect double Italian dominating functions. We check the γ d I p of some standard graphs and evaluate with γdI of such graphs. The perfect double Italian dominating functions versus perfect double Roman dominating functions are perused. The NP-completeness of this parameter is verified even when it is restricted to bipartite graphs. Finally, we characterize the graphs G of order n with γ d I p ( G ) ∈ { 3 , 4 , 5 , n , 2 n − 3 , 2 n − 4 , 2 n − 5 , 2 n − 6 } .