An edge e of a 3-connected graph G is contractible if G/e is 3-connected. A graph G is minimally 3-connected if G is 3-connected and for every \(e\in E(G)\) , \(G \backslash e\) is not 3-connected. We show that if G is a minimally 3-connected graph, then every component of the graph spanned by all non-contractible edges is either a triangle or a star. Consequently, we obtain the following results on minimally 3-connected graphs. 1. If G has a spanning tree that contains no contractible edges, then G must be one of the wheel graphs. 2. Assume that G is not a wheel graph. If G has a spanning tree that contains exactly one contractible edge, then G belongs to one of the five infinitely classes of graphs; a precise structure is given for each class. 3. The number of contractible edges of G is at least the number of edges of G minus the number of degree 3 vertices. This bound is the best possible as shown by examples from the introduction section, thus improving other established bounds on the number of contractible edges.
