Geodesic
Criterion for the existence of such a cycle that vertices beyond this cycle are independent
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part XII, Tome 497 (2020), pp. 53-79

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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. 
@article{ZNSL_2020_497_a2,
     author = {N. A. Karol'},
     title = {Criterion for the existence of such a cycle that vertices beyond this cycle are independent},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {53--79},
     publisher = {mathdoc},
     volume = {497},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2020_497_a2/}
}
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N. A. Karol'. Criterion for the existence of such a cycle that vertices beyond this cycle are independent. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part XII, Tome 497 (2020), pp. 53-79. http://geodesic.mathdoc.fr/item/ZNSL_2020_497_a2/