From a theorem of W. Mader [“Uber minimal n-fach zusammenhangende unendliche Graphen und ein Extremal problem,” Arch. Mat., Vol. 23 (1972), pp. 553–560] it follows that a k-connected (k-edge-connected) graph G = (V,E) always contains a k-connected (k-edge-connected) subgraph G′ = (V,E′) with O(k|V|) edges. T. Nishizeki and S. Poljak “K-Connectivity and Decomposition of Graphs into Forests,” Discrete Applied Mathematics, submitted) showed how G′ can be constructed as the union of k forests. H. Nagamochi and T. Ibaraki [A Linear Time Algorithm for Finding a Sparse k-Connected Spanning Subgraph of a k-Connected Graph, Algorithmica, Vol. 7 (1992), pp. 583–596] constructed such a subgraph Gk in linear time and showed for any pair x,y of nodes that Gk contains k openly disjoint (edge-disjoint) paths connecting x and y if G contains k openly disjoint (edge-disjoint) paths connecting x and y (even if G is not k-connected (k-edge-connected)). In this article we provide a much shorter proof of a common generalization of the edge- and node-connectivity versions showing that the subgraph Gk has a certain mixed connectivity property. © 1993 John Wiley & Sons, Inc.
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