Super-connectivity and super-edge-connectivity for some interconnection networks

Let G 1⁄4 ðV ;EÞ be a k-regular graph with connectivity j and edge connectivity k. G is maximum connected if j 1⁄4 k, and G is maximum edge connected if k 1⁄4 k. Moreover, G is super-connected if it is a complete graph, or it is maximum connected and every minimum vertex cut is fxjðv; xÞ 2 Eg for some vertex v 2 V ; and G is super-edge-connected if it is maximum edge connected and every minimum edge disconnecting set is fðv; xÞjðv; xÞ 2 Eg for some vertex v 2 V . In this paper, we present three schemes for constructing graphs that are super-connected and super-edge-connected. Applying these construction schemes, we can easily discuss the super-connected property and the superedge-connected property of hypercubes, twisted cubes, crossed cubes, m€ obius cubes, split-stars, and recursive circulant graphs. 2002 Elsevier Science Inc. All rights reserved.