Directional Routing via Generalized st-Numberings

We present a mathematical model for network routing based on generating paths in a consistent direction. Our development is based on an algebraic and geometric framework for defining a directional coordinate system for real vector spaces. Our model, which generalizes graph st-numberings, is based on mapping the nodes of a network to points in multidimensional space and ensures that the paths generated in different directions from the same source are node-disjoint. Such directional embeddings encode the global disjoint path structure with very simple local information. We prove that all 3-connected graphs have 3-directional embeddings in the plane so that each node outside a set of extreme nodes has a neighbor in each of the three directional regions defined in the plane. We conjecture that the result generalizes to k-connected graphs. We also show that a directed acyclic graph (dag) that is k-connected to a set of sinks has a k-directional embedding in (k − 1)-space with the sink set as the extreme nodes.