Out of the enormous branches of mathematics, graph theory plays a vital role in the fields of applied mathematics and scientific computing. One of the major fields of graph theory is the study of graph labeling problems. The present work involves the graceful labeling of a cycle of length three by joining isomorphic copies of paths to all of its vertices. The present work also analyzes the design of graceful graphs, which are created by joining caterpillars with either one or two end vertices of a cycle. Further, it establishes graceful labeling by joining two hairy cycle graphs with a common edge. To find all the results in this paper, the inverse transformation technique is used.