Geodesics in Transitive Graphs

Abstract

Let P be a double ray in an infinite graph X , and let d and d P denote the distance functions in X and in P respectively. One calls P a geodesic if d ( x , y )= d P ( x , y ), for all vertices x and y in P . We give situations when every edge of a graph belongs to a geodesic or a half-geodesic. Furthermore, we show the existence of geodesics in infinite locally-finite transitive graphs with polynomial growth which are left invariant (set-wise) under “translating” automorphisms. As the main result, we show that an infinite, locally-finite, transitive, 1-ended graph with polynomial growth is planar if and only if the complement of every geodesic has exactly two infinite components.