 
Summary: DIRECTIONAL ROUTING VIA GENERALIZED stNUMBERINGS
FRED S. ANNEXSTEIN AND KENNETH A. BERMAN
SIAM J. DISCRETE MATH. c 2000 Society for Industrial and Applied Mathematics
Vol. 13, No. 2, pp. 268279
Abstract. 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
stnumberings, 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 nodedisjoint. Such
directional embeddings encode the global disjoint path structure with very simple local information.
We prove that all 3connected graphs have 3directional 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 kconnected graphs. We also show that a
directed acyclic graph (dag) that is kconnected to a set of sinks has a kdirectional embedding in
(k  1)space with the sink set as the extreme nodes.
Key words. graph connectivity, network routing, stnumbering, matchings
AMS subject classifications. 68R10, 05C40, 68R10
PII. S0895480198333290
1. Introduction. A fundamental problem in network routing is the generation
of communication paths from a set of source nodes Y to a set of sink nodes X. Routing
