This analysis reports variations in perfect italian domination numbers across graphs and their complements, highlighting key inequalities.
Perfect Italian Domination is a type of vertex domination which can also be viewed as a graph labelling problem. The vertices of a graph \(G\) are labelled by 0, 1 or 2 in such a way that a vertex labelled 0 should have a neighbourhood with exactly two vertices in it labelled 1 each or with exactly one vertex labelled 2. The remaining vertices in the neighbourhood of the vertex labelled 0 should be all 0's. The minimum sum of all labels of the graph G satisfying these conditions is called its Perfect Italian domination number. We study the behaviour of graph complements and how the Perfect Italian Domination number varies between a graph and its complement. The Nordhaus–Gaddum type inequalities in the Perfect Italian Domination number are also discussed.
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Poovathingal et al. (2025) studied this question.
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