Decomposing Ends of Locally Finite Graphs

An important invariant of translations of infinite locally finite graphs is that of a direction as introduced by Halin . This invariant gives not much information if the translation is not a proper one. A new refined concept of directions is investigated. A double ray D of a graph X is said to be metric, if the distance metrics in D and X on V ( D ) are equivalent. It is called geodesic, if these metrics are equal. The translations leaving some metric double ray invariant are characterized. Using a result of Polat and Watkins , we characterize the translations leaving some geodesic double ray invariant.