# Highly fault-tolerant routings and diameter vulnerability for generalized hypercube graphs

## Abstract

Consider a communication network *G* in which a limited number of link and/or node faults *F* might occur. A routing ρ 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, ρ)/F*, where the surviving route graph *R(G, ρ)/F* is a directed graph consisting of all nonfaulty nodes in *G* with a directed edge from *x* to *y* iff there are no faults on the route from *x* to *y*, could be one of the fault-tolerant measures for the routing *ρ*. In this paper, we show that we can construct efficient and highly fault-tolerant routings on a *k*-dimensional generalized *d*-hypercube *C(d, k)* such that the diameter of the surviving route graph is bounded by constant for the case that the number of faults exceeds the connectivity of *C(d, k)*.

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