Criterion for the existence of such a cycle that vertices beyond this cycle are independent

This paper contains a criterion for the existence of such a cycle that the vertices beyond this cycle are independent in terms of minimum vertex degree. More specifically, if $G$ is a $2$-connected graph, $v(G) = n$ and $\delta(G) \geq \frac{n + 2}{3}$, then $G$ has a cycle such that vertices beyond this cycle are independent.