Given a graph, removing pendant vertices (vertices with only one neighbor) and vertices that have a twin (another vertex that has the same neighbors) until it is not possible yields a reduced graph, called the “pruned graph”. In this paper, we present an algorithm which computes this “pruned graph” either in linear time or in linear space. In order to achieve these complexity bounds, we introduce a data structure based on digital search trees. Originally designed to store a family of sets and to test efficiently equalities of sets after the removal of some elements, this data structure finds interesting applications in graph algorithmics. For instance, the computation of the “pruned graph” provides a new and simply implementable algorithm for the recognition of distance-hereditary graphs, and we improve the complexity bounds for the complete bipartite cover problem on bipartite distance-hereditary graphs.