  Boundary value problems for Sturm-Liouville operators with potentials from
the class $W_2^{-1}$ on a star-shaped graph are considered. We assume that the
potentials are known on all the edges of the graph except two, and show that
the potentials on the remaining edges can be constructed by fractional parts of
two spectra. A uniqueness theorem is proved, and an algorithm for the
constructive solution of the partial inverse problem is provided. The main
ingredient of the proofs is the Riesz-basis property of specially constructed
systems of functions.
