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 [InlineMediaObject not available: see fulltext.] vertices that are not incident to contractible edges. In this paper, we characterize all simple 3-connected graphs with exactly [InlineMediaObject not available: see fulltext.] 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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