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An Extremal Problem on Contractible Edges in 3-Connected Graphs

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Abstract

An edge e in a 3-connected graph G is contractible if the contraction G/e is still 3-connected. The existence of contractible edges is a very useful induction tool. Let G be a simple 3-connected graph with at least five vertices. Wu [7] proved that G has at most vertices that are not incident to contractible edges. In this paper, we characterize all simple 3-connected graphs with exactly vertices that are not incident to contractible edges. We show that all such graphs can be constructed from either a single vertex or a 3-edge-connected graph (multiple edges are allowed, but loops are not allowed) by a simple graph operation.

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References

  1. Ando, K., Enomoto, H., Saito, A., Contractible edges in 3–connected graphs. J. Combin. Theory Ser. B. 42, 87–93 (1987)

    Google Scholar 

  2. Halin, R. Untersuchungen über minimale n-fach zusammenhangende graphen. Math. Ann. 182, 175–188 (1969)

    Google Scholar 

  3. Kriesell, M.: Contractible subgraphs in 3-connected graphs. J. Combin. Theory Ser. B. 80, 32–48 (2000)

    Google Scholar 

  4. McCuaig, W.: Edge contractions in 3-connected graphs. Ars Combinatoria. 29, 299–308 (1990)

    Google Scholar 

  5. Thomassen, C.: Planarity and duality of finite and infinite graphs. J. Combin. Theory Ser. B. 29, 244–271 (1980)

    Google Scholar 

  6. Tutte, W.T.: A theory of 3–connected graphs. Nederl. Akad. Wetensch. Proc. Ser. A. 64, 441–455 (1961)

  7. Wu, H.: Contractible elements in graphs and matroids. Combinatorics, Probability and Computing. 12, 457–465 (2003)

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Authors and Affiliations

  1. Department of Mathematics, The University of Mississippi, University, MS, 38677

    Joe Anderson & Haidong Wu

Authors
  1. Joe Anderson
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  2. Haidong Wu
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Research partially supported by an ONR grant under grant number N00014-01-1-0917

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Anderson, J., Wu, H. An Extremal Problem on Contractible Edges in 3-Connected Graphs. Graphs and Combinatorics 22, 145–152 (2006). https://doi.org/10.1007/s00373-006-0661-4

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  • DOI: https://doi.org/10.1007/s00373-006-0661-4

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