Perfect Italian domination in graphs: Complexity and algorithms

Abstract

An Italian dominating function on a simple undirected graph G is a function f : V ( G ) ⟶ { 0 , 1 , 2 } satisfying the condition that for each vertex v with f ( v ) = 0 , ∑ u ∈ N G ( v ) f ( u ) ≥ 2 . An Italian dominating function f on G is called a perfect Italian dominating function on G if for each vertex v with f ( v ) = 0 , ∑ u ∈ N G ( v ) f ( u ) = 2 . The weight of a function f on a graph G , denoted by w ( f ) , is the sum ∑ v ∈ V ( G ) f ( v ) . For a simple undirected graph G , Min-PIDF is the problem of finding the minimum weight of a perfect Italian dominating function on G . First, we discuss the complexity difference between Min-PIDF and the problem of finding the minimum weight of an Italian dominating function. We then establish the NP -completeness of the decision version of Min-PIDF in chordal graphs and investigate the hardness of approximation of Min-PIDF in general graphs. Finally, we present linear time algorithms for computing the minimum weight of a perfect Italian dominating function in block graphs and series-parallel graphs.