Fault-tolerant graphs for hypercubes and tori

Motivated by the design of fault-tolerant multiprocessor interconnection networks the paper considers the following problem: given a positive integer t and graph H, construct a graph G from H by adding a minimum number Δ(t,H) of edges such that even after deleting any t edges from G the remaining graph contains H as a subgraph. We estimate Δ(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 Q _N and D_N, respectively, we show among others, that Δ(t,Q_N)=O(tNlog(logN/t+log2e)) for any t and N (t⩾2), and Δ(1,D_N)=N/2 if N is even