In this paper, we consider an arbitrary class ${\cal H}$ of rooted graphs
such that each biconnected component is
given by a representation with reflectional symmetry,
which allows a rooted graph to have several different
representations, called embeddings.
We give a general framework to design
of algorithms for enumerating embeddings of all graphs in ${\cal H}$
without repetition.
The framework yields an efficient enumeration algorithm
for a class ${\cal H}$ if the class
${\cal B}$ of biconnected graphs used in the graphs in ${\cal H}$
admits an efficient enumeration algorithm.
For example, for the class ${\cal B}$ of rooted cycles,
we can easily design an algorithm of enumerating rooted cycles
so that delivers
the difference between two consecutive cycles
in constant time in a series of all outputs.
Hence our framework implies that, for the class ${\cal H}$
of all rooted cacti, there is an algorithm that
enumerates each cactus in constant time.