Geodesic
On critically 3-connected graphs with exactly two vertices of degree~3. Part~2
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part X, Tome 475 (2018), pp. 137-173

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A graph $G$ is critically $3$-connected, if $G$ is $3$-connected and for any vertex $v\in V(G)$ the graph $G-v$ isn't $3$-connected. R. C. Entringer and P. J. Slater proved that any critically $3$-connected graph contains at least two vertices of degree $3$. In the previous paper we classify all such graphs with one additional condition: two vertices of degree $3$ are adjacent. In this paper we will consider the case of nonadjacent vertices of degree $3$. 
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     author = {A. V. Pastor},
     title = {On critically 3-connected graphs with exactly two vertices of degree~3. {Part~2}},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {137--173},
     publisher = {mathdoc},
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     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2018_475_a6/}
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A. V. Pastor. On critically 3-connected graphs with exactly two vertices of degree~3. Part~2. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part X, Tome 475 (2018), pp. 137-173. http://geodesic.mathdoc.fr/item/ZNSL_2018_475_a6/