Given a connected graph $G$ rooted with a designated vertex
or a biconnected component,
the descendants of a cut-vertex $v$
induces a connected subgraph rooted with the vertex $v$.
We consider the set ${\cal R}$ of all such rooted subgraphs in $G$, and
assign an integer, called an index,
to each of the subgraphs so that
two rooted subgraphs in ${\cal R}$ receive the same index if and only if
they are isomorphic, where their roots correspond each other.
In this paper,
assuming a procedure for computing
a signature of each graph in a class ${\cal G}$ of biconnected graphs,
we present a framework for computing indices to all rooted subgraphs
of a rooted graph which is composed of
biconnected components from ${\cal G}$.
With this framework, we can find indices to all rooted subgraphs
of a rooted outerplanar graphs in linear time.