In this paper, we present new upper bounds for the global domination andRoman domination numbers and also prove that these results are asymptoticallybest possible. Moreover, we give upper bounds for the restrained domination andtotal restrained domination numbers for large classes of graphs, and show that,for almost all graphs, the restrained domination number is equal to thedomination number, and the total restrained domination number is equal to thetotal domination number. A number of open problems are posed.
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