$G$ is a simple $r$-connected ($r \geq 1$) graph, with even number of vertices. Assume that $G$ doesn't contain $K_{1,r+1}$ as an induced subgraph. Prove that $G$ has a perfect matching.
Now, I can see why it's true without the condition on $K_{1,r+1}$: obviously, $\delta(G) \geq r$, and then, by using Tutte's theorem in a fashion of a proof of Petersen theorem ($ro(V[G-S]) \leq \sum{M_i} \leq \sum_{v \in S}d(v) \leq r|S|$, when $M_i$ is number of edges between $S$ and a component $C_i$), I'm getting to what I was needed to prove.
Am I missing something?
Thanks.