Properly 3-contractible edges in a minimally 3-connected graph

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Let G be a 3-connected graph. An edge e such that G−e is 3-connected is called a “3-removable edge”, and G is said to be “minimally 3-connected” if G has no 3-removable edge. An edge e of G is said to be a “3-contractible edge” if its contraction results in a 3-connected graph. A 3-contractible edge is said to be “properly 3-contractible” if it is contained in no triangle. Tutte's wheel theorem says that every minimally 3-connected graph with no properly 3-contractible edge is a wheel. Let E⁎(G) denote the set of edges of G which are contained in no triangle and let Ec⁎(G) denote the set of properly 3-contractible edges of G. Let U(i)(G) denote the set of degree 3 vertices x of G such that the subgraph of G induced by the neighborhood of x has i edges.In this paper, we prove that for a minimally 3-connected graph G, the following conditions (1), (2) and (3) are equivalent: (1) G is a wheel, (2) E⁎(G)=∅ and (3) U(0)(G)=U(1)(G)=∅, and that for a minimally 3-connected graph G, (I) we have |Ec⁎(G)|≥12|U(1)(G)|+|U(0)(G)|, and (II) we have |Ec⁎(G)|≥12(|E⁎(G)|+3) if G is not a wheel.We show the sharpness of the inequalities (I) and (II).