Contractible edges in 3-connected graphs

By a graph, we mean a finite undirected simple graph with no loops and no multiple edges. For a graph G and an edge e of G, we let G/e denote the graph obtained from G by contracting e (and replacing each pair of the resulting double edges by a simple edge). Let k ≥ 2 be an integer, and let G be a k-connected graph. An edge e of G is said to be k-contractible if G/e is k-connected. The set of k-contractible edges of G is denoted by Ec(G). We say that G is contraction-critically k-connected if Ec(G) = ∅. In this talk, I will survey results concerning k-contractible edges. In the study of the existence of k-contractible edges, there is a big difference between the case where 2 ≤ k ≤ 3 and the case where k ≥ 4. In the case where 2 ≤ k ≤ 3, the complete graph of order k + 1 is the only contraction-critically kconnected graph, while for k ≥ 4, there exist infinitely many contraction-critically k-connected graphs. We first consider the case where k = 4. The following result is well-known.