# Sorting by Reversals and the Theory of 4-Regular Graphs

## TL;DR

This paper connects the theory of sorting by reversals with 4-regular graph circuit partitions, revealing new relationships and extending the framework to include DCJ operations and gene assembly in ciliates.

## Contribution

It establishes a novel link between sorting by reversals and 4-regular graph theory, and explores generalizations involving DCJ operations and gene assembly.

## Key findings

- The theory of sorting by reversals is integrated into 4-regular graph circuit partition theory.
- A generalization involving double-cut-and-join (DCJ) operations is proposed.
- Connections between gene assembly in ciliates and sorting by reversals are elucidated.

## Abstract

We show that the theory of sorting by reversals fits into the well-established theory of circuit partitions of 4-regular multigraphs (which also involves the combinatorial structures of circle graphs and delta-matroids). In this way, we expose strong connections between the two theories that have not been fully appreciated before. We also discuss a generalization of sorting by reversals involving the double-cut-and-join (DCJ) operation. Finally, we also show that the theory of sorting by reversals is closely related to that of gene assembly in ciliates.