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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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