Given a graph \(G=(V,E)\) , the Perfect Italian domination function is a mapping \(f:V\rightarrow \{0,1,2\}\) such that for any vertex \(v\in V\) with f(v) equals zero, \(\sum _{u\in N(v)}f(u)\) must be two. In simpler terms, for each vertex v labeled zero, one of the following conditions must be satisfied: (1) exactly two neighbours of v are labeled 1, and every other neighbour of v is labeled zero, (2) exactly one neighbour of v is labeled 2, and every other neighbour of v is labeled zero. The weight of the function f is calculated as the sum of f(u) over all \(u\in V\) . The Perfect Italian domination problem involves finding a Perfect Italian domination function that minimizes the weight. We have devised a linear-time algorithm to solve this problem for \(P_4\) -sparse graphs, which represent well-established generalization of cographs. Furthermore, we have proved that the problem is efficiently solvable for distance-hereditary graphs. We have also shown that the decision version of the problem is NP-complete for 5-regular graphs and comb convex bipartite graphs.
