Highly Fault-Tolerant Routings in Some Cartesian Product Digraphs

Consider a communication network G in which a limited number of edge(arc) and/or vertex faults F might occur. A routing ρ, i.e. a fixed path between each pair of vertices, for the network must be chosen without knowing which components might become faulty. The diameter of the surviving route graph R(G, ρ)/F , where R(G, ρ)/F is a digraph with the same vertices as G − F and a vertex x being adjacent to another vertex y if and only if ρ(x, y) avoids F , could be an important measurement for the routing ρ. In this paper, the authors consider the Cartesian product digraphs whose factors satisfy some given conditions and show that the diameter of the surviving route graph is bounded by three for any minimal routing ρ when the number of faults is less than some integer. This result is also useful for the Cartesian product graphs and generalizes some known