# Efficient fault-tolerant routings in networks

## Abstract

We analyze the problem of constructing a network with a given number of nodes which has a fixed routing and which is highly fault tolerant. A construction is presented which forms a "product route graph" from two or more constituent "route graphs." The analysis involves the surviving route graph, which consists of all nonfaulty nodes in the network with two nodes being connected by a directed edge iff the route from the first to the second is still intact after a set of component failures. The diameter of the surviving route graph is a measure of the worst-case performance degradation caused by the faults. The number of faults tolerated, the diameter, and the degree of the product graph are related in a simple way to the corresponding parameters of the constituent graphs. In addition, there is a "padding theorem" which allows one to add nodes to a graph and to extend a previous routing. © 1987.