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Let $G$ be a graph with vertex set $V(G)$. A function $f:V(G)\rightarrow\{0,1,2\}$ is a Roman dominating function on $G$ if every vertex $v\in V(G)$for which $f(v)=0$ is adjacent to at least one vertex $u\in V(G)$ such that$f(u)=2$. The Roman domination number of $G$ is the minimum weight$\omega(f)=\sum_{x\in V(G)}f(x)$ among all Roman dominating functions $f$ on$G$. In this article we study the Roman domination number of direct productgraphs and rooted product graphs. Specifically, we give several tight lower andupper bounds for the Roman domination number of direct product graphs involvingsome parameters of the factors, which include the domination, (total) Romandomination, and packing numbers among others. On the other hand, we prove thatthe Roman domination number of rooted product graphs can attain only threepossible values, which depend on the order, the domination, and the Romandomination numbers of the factors in the product. In addition, theoreticalcharacterizations of the classes of rooted product graphs achieving each ofthese three possible values are given.

Roman domination in direct product graphs and rooted product graphs