Geodesic
Restriction on minimum degree in the contractible sets problem
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part XIII, Tome 518 (2022), pp. 114-123

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Let $G$ be a $3$-connected graph. A set $W \subset V(G)$ is contractible if $G(W)$ is connected and $G - W$ is a $2$-connected graph. In 1994, McCuaig and Ota formulated the conjecture that, for any $k \in \mathbb{N}$, there exists $m \in \mathbb{N}$ such that any $3$-connected graph $G$ with $v(G) \geqslant m$ has a $k$-vertex contractible set. In this paper we prove that, for any $k \geqslant 5$, the assertion of the conjecture holds if $\delta(G) \geqslant \left[ \frac{2k + 1}{3} \right] + 2$. 
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     author = {N. A. Karol},
     title = {Restriction on minimum degree in the contractible sets problem},
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N. A. Karol. Restriction on minimum degree in the contractible sets problem. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part XIII, Tome 518 (2022), pp. 114-123. http://geodesic.mathdoc.fr/item/ZNSL_2022_518_a2/