Minimal and bidirectional fault-tolerant routings for k-hypercube graphs

A communication network G is considered in which a limited number of link and/or node faults F might occur. A routing rho for the network (a fixed path between each pair of nodes) must be chosen without knowing which components might become faulty. The diameter of the surviving route graph R(G, rho )/F (denoted by (D(R(G, rho )/F))), where two nonfaulty nodes are connected by an edge if there are no faults on the route between them, could be one of the fault-tolerant measures for the routing rho . It is shown that for any k(>or=3), there exists a bidirectional and minimal routing (i.e. a routing in which the route between any two nodes (if defined) is assigned to one of the shortest paths between them, and is the same in both directions) lambda /sub k/ on the k-dimensional hypercube graph C/sub k/ such that D(R(C/sub k/, lambda /sub k/)/F)>