Roman, Italian, and 2-Domination

Abstract

Motivated by finding a way to connect the Roman domination number and 2-domination number, which are in general not comparable, we consider a parameter called the Italian domination number (also known as the Roman $(2)$-domination number). This parameter is bounded above by each of the other two. Bounds on the Italian domination number in terms of the order of the graph are shown. The value of the Italian domination number is studied for several classes of graphs. We also compare the Italian domination number with the 2-domination number.