Routed planar networks

Abstract

Modeling a road network as a planar graph seems very natural. However, in studying continuum limits of such networks it is useful to take {\em routes} rather than {\em edges} as primitives. This article is intended to introduce the relevant (discrete setting) notion of {\em routed network} to graph theorists. We give a naive classification of all 71 topologically different such networks on 4 leaves, and pose a variety of challenging research questions.