# Finding Balance: Split Graphs and Related Classes

## TL;DR

This paper explores split graphs and their unbalanced variants, establishing bijections with related combinatorial structures like set covers, bipartite graphs, and posets, extending previous counting results to new settings.

## Contribution

It generalizes the concept of unbalanced split graphs to various combinatorial classes and establishes bijections, broadening understanding of their enumeration and structure.

## Key findings

- Unbalanced split graphs correspond bijectively to smaller split graphs.
- The concepts extend to minimal set covers, bipartite graphs, and posets.
- New counting and structural insights are provided for these classes.

## Abstract

A graph is a split graph if its vertex set can be partitioned into a clique and a stable set. A split graph is unbalanced if there exist two such partitions that are distinct. Cheng, Collins and Trenk (2016), discovered the following interesting counting fact: unlabeled, unbalanced split graphs on $n$ vertices can be placed into a bijection with all unlabeled split graphs on $n-1$ or fewer vertices. In this paper we translate these concepts and the theorem to different combinatorial settings: minimal set covers, bipartite graphs with a distinguished block and posets of height one.