Abstract
We prove that if G is highly connected, then either G contains a non-separating connected subgraph of order three or else G contains a small obstruction for the above conclusion. More precisely, we prove that if G is k-connected (with k ≥ 2), then G contains either a connected subgraph of order three whose contraction results in a k-connected graph (i.e., keeps the connectivity) or a subdivision of \({K_4^-}\) whose order is at most 6.
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Shinya Fujita’s work is supported by the JSPS Research Fellowships for Young Scientists. Ken-ichi Kawarabayashi’s 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 Small Subgraphs in k-connected Graphs. Graphs and Combinatorics 26, 499–511 (2010). https://doi.org/10.1007/s00373-010-0930-0
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DOI: https://doi.org/10.1007/s00373-010-0930-0