A family of sets is called a star if there exists an element (a center) in each of its sets. Given a graph G and an integer r⩾1, let Ir(G) be the family of independent sets of size r of G. A star S of Ir(G) is maximum if no star of Ir(G) is bigger than S. In 2011, Hurlbert and Kamat conjectured that in trees there exists a maximum star of Ir(G) that is centered at a leaf for any r, and they also showed that this conjecture is true for r⩽4. In 2021, Estrugo and Pastine proposed the following problem: Find which leaves are the centers of the maximum stars in caterpillars. In this contribution, we consider the problem on the maximum star of Ir(G) for a connected graph G with at least one leaf. Firstly, we show that in unicyclic graphs with at least one leaf there exists a maximum star of Ir(G) that is centered at a leaf for r⩽4, and there exists a counterexample in which the maximum star of Ir(G) is not centered at a leaf for r⩾5. Secondly, we show that in connected graphs having at least one leaf and at least two cycles, there exist two counterexamples in which the maximum stars of Ir(G) are not centered at leaves for r⩾3. Finally, we determine the centers of the maximum stars of Ir(G) for three types of caterpillars, which partially solves the problem proposed by Estrugo and Pastine.