  Roman domination is one of the many variants of domination that keeps most of
the complexity features of the classical domination problem. We prove that
Roman domination behaves differently in two aspects: enumeration and extension.
We develop non-trivial enumeration algorithms for minimal Roman domination
functions with polynomial delay and polynomial space. Recall that the existence
of a similar enumeration result for minimal dominating sets is open for
decades. Our result is based on a polynomial-time algorithm for Extension Roman
Domination: Given a graph $G = (V,E)$ and a function $f:V\to\{0,1,2\}$, is
there a minimal Roman domination function $\Tilde{f}$ with $f\leq \Tilde{f}$?
Here, $\leq$ lifts $0< 1< 2$ pointwise; minimality is understood in this order.
Our enumeration algorithm is also analyzed from an input-sensitive viewpoint,
leading to a run-time estimate of $\Oh(\RomanUpperbound^n)$ for graphs of order
n; this is complemented by a lower bound example of
$\Omega(\RomanLowerbound^n)$.
