Birational equivalence of reduced graphs

We study a canonical desingularization process for oriented reduced graphs. We use it to give an arithmetical characterization for these graphs by means of sequences of natural numbers, based on the representation of a partial ordering by its maximal chains. For such graphs we define, in analogy with Algebraic Geometry, similar tools and language as in birational geometry. We study a class of birationally equivalent graphs giving it a graph structure and describing, in an explicit way, a canonical graph in the class with a minimal number of points. (C) 1998 Elsevier Science B.V.