Geodesic
Decomposition of a~$2$-connected graph into three connected subgraphs
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part IX, Tome 464 (2017), pp. 26-47

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Let $G$ be a $2$-connected graph on $n$ vertices such that any its $2$-vertex cutset splits $G$ into at most three parts and $n_1+n_2 +n_3=n$. We prove that there exists a decomposition of the vertex set of $G$ into three disjoint subsets $V_1$, $V_2$, $V_3$, such that $|V_i|=n_i$ and the induced subgraph $G(V_i)$ is connected for each $i$. 
@article{ZNSL_2017_464_a1,
     author = {D. V. Karpov},
     title = {Decomposition of a~$2$-connected graph into three connected subgraphs},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {26--47},
     publisher = {mathdoc},
     volume = {464},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_464_a1/}
}
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D. V. Karpov. Decomposition of a~$2$-connected graph into three connected subgraphs. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part IX, Tome 464 (2017), pp. 26-47. http://geodesic.mathdoc.fr/item/ZNSL_2017_464_a1/