On Some Graphs Whose Domination Number Is the Perfect Italian Domination Number

Perfect Italian Domination (PID) is a vertex labelling of a graph G by numbers from the set \(\{0,1,2\}\) such that a vertex in G labelled 0 has a neighbourhood where the summation of the labels of the vertices in it is precisely 2. The summation of labels on the vertices of the graph which satisfy the PID labelling is known as its PID number, and \(\gamma _I^p(G)\) is the minimum possible PID number of a graph G. We find some characterization of graphs for which \(\gamma (G)=\gamma _I^p(G)\) . We also find a lower bound for |V(G)|, which satisfies the same. Further, we discuss the graphs that satisfies \(\gamma (G)=\gamma _I^p(G)=2\) or \(\frac{n}{2}\) . A realisation problem is used to prove that PID cannot be bounded by a scalar multiple of the Domination number.