1 arXiv:1108.3351v2 [math.CO] 18 Jan 2012 An Alternative Construction to the Transitive Closure of a Directed Graph Kenneth L. Price October 30, 2018 University of Wisconsin Oshkosh One must add arrows which are forced by transitivity to form the transitive closure of a directed graph. We introduce a construction of a transitive directed graph which is formed by adding vertices instead of arrows and which preserves the transitive relationships formed by distinct vertices in the original directed graph. Our construction does not apply to all directed graphs. We start by fixing notation and defining compression maps in Section 1. The motivation for this investigation comes from algebra through connections with inci- dence rings, which are rings of functions defined on sets with relations. None of the algebra applications are covered here, but we mention anyway a good reference for incidence rings is [4]. The class of generalized incidence rings was introduced by G. Abrams in [1]. His construction is based on the balanced relations. In Section 2 we Abstract