Inverse nodal problems consist in constructing operators from the given zeros of their eigenfunctions. In this work, we deal with the inverse nodal problems of reconstructing the Sturm-Liouville operator on a star graph with δ'(subscript s) couplings at the central vertex. The uniqueness theorem is proved and a constructive procedure for the solution is provided from a dense subset of zeros of the eigenfunctions for the problem as a data.