NEW RESULTS IN GRAPH ROUTING

Consider a communication network or an undirected graph G in which a limited number of link and/or node faults F might occur. A routing ρ for the network (at most one path, called a route, for each ordered pair of nodes) must be chosen without knowing which components might be faulty. A routing is said to be minimal if any route from x to y is assigned to one of the shortest paths from x to y, and is said to be bidirectional if for any ordered pair (x, y) the route from x to y and the route from y to x are assigned to the same path. The diameter of the surviving route graph R(G, ρ)/F (denoted by D(R(G, ρ)/F)), where two nonfaulty nodes x and y are connected by a directed edge if 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 there exists a bidirectional and minimal routing λk on a k-dimensional hypercube graph Ck such that D(R(Ck, λk)F) ≤ 2 for any set of faults F (|F| < k) in the case that k = 3m and k = 3m + 1 (m ≥ 1), and that there exists a bidirectional and almost minimal routing π3m+2 (m ≥ 0) on C3m+2 such that D(R(C3m+2, π3m+2)/F) ≤ 2 for any set of faults F (|F| < 3m + 2). These are solutions for the open problem raised by Dolev et al. (1987, Inform. and Comput.72, 180-196). We also show that we can construct a routing ρ for any graph G in some class of (k + 1)-node connected graphs such that D(R(G, ρ)/F) ≤ 2 for any set of faults F (|F| ≤ k). As long as faults are assumed to occur in a network, the diameter of the surviving route graph for the network is more than one. Thus, the routing shown here is best possible and is said to be optimal. © 1993 Academic Press, Inc.