Partition Coe cients of Acyclic Graphs ?

We develop the concept of a \closure space" which appears with diierent names in many aspects of graph theory. We show that acyclic graphs can be almost characterized by the partition coeecients of their associated closure spaces. The resulting nearly total ordering of all acyclic graphs (or partial orders) provides an eeective isomorphism lter and the basis for eecient retrieval in secondary storage. 1 Binary Partitions In this paper we combine two mathematical threads and apply them in a graph-theoretic context. The rst thread of binary partitions was studied by Euler as early as 1750. The second thread involving closure spaces is of more recent origin. A binary partition of a positive integer N is its expression as a sum of powers of 2. Mahler 16], and Churchhouse 3] 4] have studied binary partitions from a number theoretic point of view. Because our intention is to connect these partitions with closure spaces, we will connne our attention to the special case where N is also a power of 2. By a binary partition of 2