Vertex Removal on Perfect Italian Domination and $\gamma_I^p$-Stability of Graphs

Perfect Italian Domination is a domination concept where all vertices are assigned one of the labels among $0$, $1$ and $2$ such that the sum of the labels in the neighbourhood of every vertex labelled $0$ should be exactly $2$. If the zero-labelled vertices are adjacent to any other vertex, they should all be zero-labelled. We examine a few graph classes and discuss in detail the criticality concept of Perfect Italian Domination. We also define $\gamma_I^p-$ stable graphs and PID critical graphs. Following our definitions of $\gamma_I^p$-stable and PID critical graphs, we have grouped some graph classes. We characterise a family of trees that is $\gamma_I^p$-stable.