Roman domination is one of few examples where the related extension problem is polynomial-time solvable even if the original decision problem is NP-complete. This is interesting as it allows to establish polynomial-delay enumeration algorithms for finding minimal Roman dominating functions, while it is open for more than four decades if all minimal dominating sets can be enumerated efficiently. To find the reason why this is the case, we combine the idea of hitting set with the idea of Roman domination. We hence obtain a new combinatorial problem, called Roman Hitting Set, generalizing Roman Domination in a natural way. This allows us to expand the frontier of polynomial-delay enumerability, as opposed to transversal-hardness. The generalization Roman Hitting Set is insofar interesting as it always allows for polynomial-delay enumeration, and we also give an explicit input-sensitive enumeration algorithm for this problem that is optimal in the sense that we can also give a matching lower bound. Based on Roman Hitting Set, we also discuss consequences for Roman variants of Vertex Cover and Feedback Vertex Set. For minimal Roman vertex covers of size upper-bounded by k, we can also deduce an optimal parameterized enumeration algorithm. Also viewed from Parameterized Complexity, the studies on extension problems are interesting, giving more examples of parameterized problems complete for \(\textsf {W}[3]\) . Finally, we also consider implicitly given hypergraphs and prove that, for the example of Roman FVS, our polynomial-delay enumeration algorithm can still be used, avoiding an explicit construction of the hypergraph.