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In this paper, we obtain a necessary condition for the Skolem gracefulness of the disjoint union of \( k \) signed stars \( K_{1,r_i}, 1 \leq i \leq k \), which we call a \( k \)-signed star \( St(r_1,r_2,\ldots,r_k) \). We also present results on the Skolem gracefulness of the 2-signed star \( St(r_1,r_2) \).