Geodesic
Every $3$-connected graph on at least $13$ vertices has a contractible set on $5$ vertices
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part XIII, Tome 518 (2022), pp. 5-93

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A subset $H$ of the set of vertices of a $3$-connected finite graph $G$ is called contractible if $G(H)$ is connected and $G - H$ is $2$-connected. We prove that every $3$-connected graph on at least $13$ vertices has a contractible set on $5$ vertices. And there is a $3$-connected graph on $12$ vertices that does not contain a contractible set on $5$ vertices. 
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     author = {N. Yu. Vlasova},
     title = {Every $3$-connected graph on at least $13$ vertices has a contractible set on $5$ vertices},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--93},
     publisher = {mathdoc},
     volume = {518},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2022_518_a0/}
}
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N. Yu. Vlasova. Every $3$-connected graph on at least $13$ vertices has a contractible set on $5$ vertices. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part XIII, Tome 518 (2022), pp. 5-93. http://geodesic.mathdoc.fr/item/ZNSL_2022_518_a0/