Math. Program. 84: 555–563 (1999) / DOI 10.1007/s10107990072b  Springer-Verlag 1999 Ervin Gy˝ ori · Tibor Jordán How to make a graph four-connected Received October 1995 / Revised version received March 1997 Published online March 16, 1999 Abstract. Two extremal-type versions of the connectivity augmentation problem are investigated. Using ear-decomposition theorems we prove that (i) every 2-connected graph on n vertices can be made 4-connected by adding at most n new edges, and (ii) every 3-connected and 3-regular graph on n ≥ 8 vertices can be made 4-connected by adding n/2 new edges. 1. Introduction A graph G = (V, E) is k-connected if |V |≥ k + 1 and deleting any set of less than k vertices results in a connected graph. By Menger’s theorem this is equivalent to the fact that there exist k pairwise vertex-disjoint paths between any two vertices of G. The connectivity augmentation problem with target connectivity k is the following: G = (V, E) G k This is a challenging unsolved problem for general k. Although the answer is known for small values of k (that is, a good characterization was found and there exist polynomial algorithms for increasing the connectivity optimally up to k = 3), it is still an open question whether the above problem can be solved in polynomial time for k ≥ 4. Further related results on the augmentation problem can be found in the survey paper [7], where its directed and edge-connectivity versions are also investigated. In this paper we solve two special cases of the of the augmentation problem. In this version the goal is to determine the maximum possible size of a smallest augmenting set in terms of the target k, the number of vertices n := |V |, and the connectivity l of the starting graph G. This kind of extremal question (for arbitrary n, k, l ) was first posed by Bollobás [2, page 49.]. So far the extremal value has been found only in some special cases. If the starting graph G has connectivity l = 0 and k ≥ 2 is arbitrary, the extremum is ⌈kn/2⌉, which follows from the well-known construction E. Gy˝ ori: Mathematical Institute of the Hungarian Academy of Sciences, Reáltanoda u. 13-15, H-1053 Budapest, Hungary T. Jordán: CWI, Kruislaan 413, 1098 SJ, Amsterdam, The Netherlands. Partially supported by OTKA grants F014919, T17580. Current address: BRICS, Department of Computer Science, University of Aarhus, DK-8000 Aarhus C, Denmark Mathematics Subject Classification (1991): 05C40, 05C35