Geodesics in transitive graphs (Q1924135)

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.