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Graceful graphs were first studied by Rosa [17]. A graceful labeling \(f\) of a graph G is a one-to-one map from the set of vertices of \(G\) to the set {0.1,., |E(G)|}. where for edges \(xy\), the induced edge labels |f(x) – f (y)| form the set {1,2,., |E(G)|, with no label repeated. In this paper, we investigate the set of labels taken by the central vertex of the star in the graph \(K_{1.m-1} \oplus C_n\), for each graceful labeling. We also study gracefulness of certain unicyclic graphs where paths \(P_3, P_2\) are pendant at vertices of the cycle. For these unicyclic graphs, the deletion of any edge of the cycle does not result in a caterpillar.