Rooted Routing in the Plane

The problem A graph G consists of a set V G of vertices and a set E G of edges each of which is an ordered pair of vertices These objects despite their simple struc ture can be used to model important properties of a wide variety of math ematical and physical systems One of their most important applications is to the study of routing in networks Here the vertices represent sites cities computers airports and the edges represent connections roads telephones ights A fundamental result in routing theory concerns disjoint paths between two speci ed sets of vertices in a graph G A path is a sequence of distinct vertices between each consecutive pair of which there is an edge The endpoints of a path are the rst and last elements of the sequence The vertices of a path of length at least three form a simple cycle if there is an edge between the path s endpoints