Roman domination in direct product graphs and rooted product graphs

Let $G$ be a graph with vertex set $V(G)$. A function $f:V(G)\rightarrow {0,1,2}$isaRomandominatingfunctionon$G$ifeveryvertex$v\in V(G)$forwhich$f(v)=0$isadjacenttoatleastonevertex$u\in V(G)$suchthat$f(u)=2$\mathrm{.}TheRomandominationnumberof$G$istheminimumweight$\omega(f)=\sum_{x\in V(G)}f(x)$amongallRomandominatingfunctions$f$on$G$. In this article we study the Roman domination number of direct product graphs and rooted product graphs. Specifically, we give several tight lower and upper bounds for the Roman domination number of direct product graphs involving some parameters of the factors, which include the domination, (total) Roman domination, and packing numbers among others. On the other hand, we prove that the Roman domination number of rooted product graphs can attain only three possible values, which depend on the order, the domination, and the Roman domination numbers of the factors in the product. In addition, theoretical characterizations of the classes of rooted product graphs achieving each of these three possible values are given.