  We suggest a new formulation of the inverse spectral problem for second-order
functional-differential operators on star-shaped graphs with global delay. The
latter means that the delay, being measured in the direction to a specific
boundary vertex, called the root, propagates through the internal vertex to
other edges. Now, we intend to recover the potentials given the spectra of two
boundary value problems on the graph with a common set of boundary conditions
at all boundary vertices except the root. We prove the uniqueness theorem and
obtain a constructive procedure for solving this inverse problem assuming that
the common boundary conditions are of the Robin type and they are pairwise
linearly independent. Although we focus on graphs with equal edges when the
delay parameter also coincides with their length, the proposed formulation is
expected to be relevant for more general situations including non-star trees
with non-equal edges and with a wide range for the global delay parameter.
