Large survivable nets and the generalized prisms

For a graph G with vertices labeled 1,2,…,n and a permutation α in Sn, the α-generalized prism over G, α(G), consists of two copies of G, say Gx and Gy, along with the edges (xi,yα(i)), for 1 ⩽ i ⩽ n. In [Discrete Appl. Math. 30 (1991) 229–233], the importance of building large graphs by using generalized prisms is indicated, and the connectivity of the generalized prisms is discussed. Letf(G) denote a graphical measure of G, and let F̄f(G) denote the maximum value of F(H) taken over all subgraphs H of G. When f is a vulnerability measure, networks G with Ff(G) = F̄f(G) would usually be regarded as survivable, for a knowledgeable enemy would find no especially attractive targets. In this note, we investigate sufficient conditions for Ff(α(G)) = F̄f(α(G)), for any αϵS¦v(G)¦, where Ff is the connectivity, edge-connectivity, or the minimum degree. As a result, we obtain a method to produce large survivable networks by repeatedly taking generalized prisms, and extend some results in [Discrete Appl. Math. 30 (1991) 229–233]. Related polynomial algorithms are discussed.