Partition coefficients of acyclic graphs

Abstract

Abstract. We develop the concept of a "closure space" which appears with different names in many aspects of graph theory. We show that acyclic graphs can be almost characterized by the partition coefficients of their associated closure spaces. The resulting nearly total ordering of all acyclic graphs (or partial orders) provides an effective isomorphism filter and the basis for efficient retrieval in secondary storage. Closure spaces and their partition coefficients provide the theoretical basis for a new computer system being developed to investigate the properties of arbitrary acyclic graphs and partial orders. Binary PartitionsIn this paper we combine two mathematical threads and apply them in a graphtheoretic context. The first 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 confine our attention to the special case where N is also a power of 2.By a binary partition of 2 n we mean a sequence of non-negative integers <.-.,ak.-.>,0<k<nsuchthat an 9 2 n + an-1 9 2 n-1 + an-2 9 2 n-2 + "" "+ al 9 21 -}-a0 9 20 = 2 n (1) or ~=0 ak 9 2 ~ --2 n. The set of all such partitions we denote by 7 ~n. (From now on we frequently omit the adjective "binary".)Several characteristics of (1) are readily apparent. First, as r 0 if and only if ak = 0 for all 0 ~ k < n. Second, since the right hand side is even and all terms ak 9 2 k, k > 0 must be even, the coefficient a0 must be even. Third, if < ...,ak,ak-1,... > is a partition, then < ...,ak -1, ak_l + 2,-.. > must be as well. And fourth, if < an,...,ak,...,ao > is a partition of 2 ~ then < as, -9 ", ak, 9 9 a0, 0 > is a partition of 2 n+l .With these observations, it is not difficult to write a process which generates all partitions in lexicographic order. Doing so, and displaying each partition,