Geodesics in Transitive Graphs

LetPbe a double ray in an infinite graphX, and letdanddPdenote the distance functions inXand inPrespectively. One callsPageodesicifd(x, y)=dP(x, y), for all verticesxandyinP. 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.