Geodesic
On contractible 5-vertex subgraphs of a 3-connected graph
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part X, Tome 475 (2018), pp. 22-40

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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 $11$ vertices with minimal degree at least $4$ has a contractible set on $5$ vertices. 
@article{ZNSL_2018_475_a1,
     author = {N. Yu. Vlasova},
     title = {On contractible  5-vertex subgraphs of a 3-connected graph},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {22--40},
     publisher = {mathdoc},
     volume = {475},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2018_475_a1/}
}
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N. Yu. Vlasova. On contractible  5-vertex subgraphs of a 3-connected graph. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part X, Tome 475 (2018), pp. 22-40. http://geodesic.mathdoc.fr/item/ZNSL_2018_475_a1/