The 3-connected graphs with a longest path containing precisely two contractible edges

Previously the authors characterized the 3-connected graphs with a Hamilton path containing only two contractible edges. In this paper we extend this result by showing that if a 3-connected graph has a diameter containing only two contractible edges, then that diameter is a Hamilton path. INTRODUCTION AND TERMINOLOGY All graphs in this paper are finite, undirected and simple. Let G be a 3-connected graph. An edge e xy in G is said to be contractible if the graph obtained from G by contracting e is also 3-connected. Otherwise, e is said to be noncontractible. For G i=K4 and e = xy E E(G), one easily sees that e is noncontractible if and only if there exists S E V (G) such that S = {x, y, s} is a 3-cutset of G; in that case we say that e and S are associates of each other. We use Ec (G) to denote the set of all contractible edges of G and En (G) for the set of all noncontractible edges. For H a subgraph of G we set Ec(H) = Ec(G) n E(H) and En(H) = En(G) n E(H). We also let G[H] denote the subgraph induced by V(H). If no confusion can arise, we will often use H for any of V(H), E(H) or the subgraph H. For x E V(G), N(x) will denote the set of neighbours of x in G. A consequence of a result in Dean, Hemminger and Toft [DHT87] is that every diameter of a 3-connected graph G contains at least two contractible edges of G. In [ACH93] the authors characterized the 3-connected graphs with a Hamilton path containing only two contractible edges; we denote this class by H2 . Now let 'D2 denote the class of 3-connected graphs G that have a diameter containing only two * Acknowledges support of New Zealand FRST ** Supported by NSF grant #INT-9221418. Austt~alasian Journal of Combinatodcs 13(1996), pp.3-13 contractible edges of G. In this paper we show that such a diameter is in fact a Hamilton path. That is, our goal is to prove the following.