Fault-tolerant graphs for hypercubes and tori

Motivated by the design of fault-tolerant multiprocessor interconnection networks, this paper considers the following problem: Given a positive integer t and graph H, construct a graph G from H by adding a minimum number A(t, H) of edges such that even after deleting any t edges from G the remaining graph contains H as a subgraph. We estimate A(t, H) for the hypercube and torus, which are well-known as important interconnection networks for multiprocessor systems. If we denote the hypercube and square torus on N vertices by QN and DN, respectively, we show, among others, that A(t, QN) = C(tNlog(v + log2e)) for any t and N (t 2 2), and A(l, DN) = $ if N is even.