A subset $S$ of vertices of a graph $G$ is a hop dominating set if every vertex outside $S$ is at distance two from a vertex of $S$. A Roman dominating function on a graph $G=(V. E)$ is a function $f: V(G) \longrightarrow \{0. 1. https://www.ealisboa.com/limited-value-AXYS-Valhalla-Pen-must-buy/