By Kuhn D., Osthus D.

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Our main result states that every 3-connected graph G has a vertex x such that G − −x is 3-connected unless G is isomorphic to K 3,3 , K 2 × K 3 , or to a wheel K 1 ∗ C for some ≥ 3. This leads to a generator theorem for 3-connected graphs in terms of series parallel extensions. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 41–54, 2008 Keywords: connectivity; reduction; generator theorem; vertex deletion; vertex suppression; series– parallel graph; separating cycle; edge adjunction; edge insertion 1.

Different upper bounds on γk (G) are known in terms of the order n and the minimum ¨ degree δ of G. In this self-contained article, we present an Erdos-type result, from which some of these bounds follow. In particular, we improve the bound γk (G) ≤ (2k − δ − 1)n/(2k − δ) for (δ + 3)/2 ≤ k ≤ δ − 1, proved by Chen and Zhou in 1998. Furthermore, we characterize the extremal graphs in the inequality γk (G) ≤ kn/(k + 1), if k ≤ δ, of Cockayne et al. This characterization generalizes that of graphs realizing γ1 (G) = γ(G) = n/2.

In this article, we study k-detour subgraphs of the n-dimensional cube, Q n , with few edges or with moderate maximum degree. Let (k, n) denote the minimum possible maximum degree of a k-detour subgraph of Q n . The main result is that for every k ≥ 2 and Contract grant sponsor: National Science Foundation (to A. ); Contract grant number: DMS-0400498. Journal of Graph Theory © 2007 Wiley Periodicals, Inc. 55 56 JOURNAL OF GRAPH THEORY n ≥ 21, n ln ln n n ≤ (k, n) ≤ 20 . ln n ln n On the other hand, for each fixed even k ≥ 4 and large n, there exists a k-detour subgraph of Q n with average degree at most 2 + 24−k/2 + o(1).