On Separating Families of Bipartitions

Abstract

In this paper, we focus on families of bipartitions, i.e. set partitions consisting of at most two components. We say that a family of bipartitions is a separating family for a set $S$ if every two elements in $S$ can be separated by some bipartition. Furthermore, we call a separating family minimal if no proper subfamily is a separating family. We characterize the set of all minimal separating families of maximum size for arbitrary set $S$ as the set of all spanning trees on $S$ and enumerate minimal separating families of maximum size. Furthermore, we enumerate separating families of arbitrary size, which need not be minimal.