Geodesic
Large Contractible Subgraphs of a 3-Connected Graph
Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 1, pp. 83-101

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Let m ≥ 5 be a positive integer and let G be a 3-connected graph on at least 2m + 1 vertices. We prove that G has a contractible set W such that m ≤ |W| ≤ 2m − 4. (Recall that a set W ⊂ V (G) of a 3-connected graph G is contractible if the graph G(W) is connected and the graph G − W is 2-connected.) A particular case for m = 4 is that any 3-connected graph on at least 11 vertices has a contractible set of 5 or 6 vertices.
Keywords: connectivity, 3-connected graph, contractible subgraph 
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Karpov, Dmitri V. Large Contractible Subgraphs of a 3-Connected Graph. Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 1, pp. 83-101. http://geodesic.mathdoc.fr/item/DMGT_2021_41_1_a5/