Friendly Domination in Graph (European Journal of Pure and Applied Mathematics, 2025)
Friendly domination combines two ideas in graph theory: domination, which captures influence, and friendly sets, which balance that influence. We study the friendly dominating set and the friendly domination number, the smallest size of a set that is both dominating and friendly. First, we prove that for every graph the usual domination number is never larger than its friendly domination number, and we identify all graphs whose friendly domination number equals one or two. We then show that the gap between these two parameters can be made arbitrarily large. Exact formulas are derived for paths and cycles. Sharp Nordhaus–Gaddum bounds are obtained for the sum and product of the friendly domination number of a graph and its complement. Finally, we give complete structural characterizations of friendly dominating sets in the join and corona of two graphs.
Keywords: Friendly set; Friendly Domination; Domination