If there are no isolated vertices, then all vertices have degree at least 1. Hence, . Now, is the maximum degree of any vertex. If has edges and , then there exists a vertex incident to every edge in . This would imply that has only one component. Why are you limiting to only connected graphs that also happen to be stars?

Instead, assume the hypothesis holds for any graph with fewer than edges. Let be a graph with edges, and let for some . Now, by the IH, satisfies the hypothesis. Add back . If the smallest integer greater than or equal to equals the smallest integer greater than or equal to , then a matching in that satisfies the hypothesis is a matching in that satisfies the hypothesis. So, figure out when adding to will make that number increase. That will help you figure out what you need to prove. (Hint: )