Abstract
We prove that if G is k-connected (with k ≥ 2), then G contains either a cycle of length 4 or a connected subgraph of order 3 whose contraction results in a k-connected graph. This immediately implies that any k-connected graph has either a cycle of length 4 or a connected subgraph of order 3 whose deletion results in a (k − 1)-connected graph.
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This work is supported by the JSPS Research Fellowships for Young Scientists.
Research partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research, by Sumitomo Foundation, by Inamori Foundation and by Kayamori Foundation.
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Fujita, S., Kawarabayashi, Ki. Contractible Triples in Highly Connected Graphs. Ann. Comb. 14, 457–465 (2010). https://doi.org/10.1007/s00026-011-0070-0
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DOI: https://doi.org/10.1007/s00026-011-0070-0