The 3-Connected Graphs with Exactly k Non-Essential Edges

Let G be a simple 3-connected graph. An edge e of G is essential if neither the deletion G e nor the contraction G/e is both simple and 3-connected. In this study, we show that all 3-connected graphs with k(k ≥ 2) non-essential edges can be obtained from a wheel by three kinds of operations which defined in the paper.