  We introduce the concept of a prime band in a string algebra $\Lambda$ and
use it to associate to $\Lambda$ its finite bridge quiver. Then we introduce a
new technique of `recursive systems' for showing that a graph map between
finite dimensional string modules lies in its stable radical. Further we study
two classes of non-domestic string algebras in terms of some connectedness
properties of its bridge quiver. `Meta-$\bigcup$-cyclic' string algebras
constitute the first class that is essentially characterized by the statement
that each finite string is a substring of a band. Extending this class we have
`meta-torsion-free' string algebras that are characterized by a dichotomy
statement for ranks of graph maps between string modules--such maps either have
finite rank or are in the stable radical. Their stable ranks can only take
values from $\{\omega,\omega+1,\omega+2\}$.
