Perfect Italian domination in graphs: Complexity and algorithms - Pradhan, D.; Banerjee, S.; Liu, Jia-Bao (Discrete Applied Mathematics, 2022)
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∈NG(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∈NG(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.