On Separating Families of Bipartitions

TL;DR

This paper characterizes and enumerates minimal and arbitrary separating families of bipartitions for a set, linking maximum size families to spanning trees and providing comprehensive counts.

Contribution

It provides a complete characterization of minimal separating families of bipartitions and establishes their correspondence with spanning trees, including enumeration results.

Findings

Minimal separating families correspond to spanning trees.

Enumeration of minimal separating families of maximum size.

Counting separating families of arbitrary sizes.

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.