On four-connecting a triconnected graph

The author considers the problem of finding a smallest set of edges whose addition four-connects a triconnected graph. This is a fundamental graph-theoretic problem that has applications in designing reliable networks. He presents an O(nα(m,n)+m) time sequential algorithm for four-connecting an undirected graph G that is triconnected by adding the smallest number of edges, where n and m are the number of vertices and edges in G, respectively, and α(m, n) is the inverse Ackermann function. He presents a new lower bound for the number of edges needed to four-connect a triconnected graph. The form of this lower bound is different from the form of the lower bound known for biconnectivity augmentation and triconnectivity augmentation. The new lower bound applies for arbitrary k, and gives a tighter lower bound than the one known earlier for the number of edges needed to k-connect a (k-1)-connect graph. For k=4, he shows that this lower bound is tight by giving an efficient algorithm for finding a set edges with the required size whose addition four-connects a triconnected graph