# Switches in Eulerian graphs

## TL;DR

This paper proves that transforming a simple graph into an Eulerian graph with minimal edge switches is NP-hard, and that converting one Eulerian graph into another via 2-switches is always possible but also NP-hard to optimize.

## Contribution

It establishes NP-hardness results for both the minimal transformation problem and the shortest sequence problem between Eulerian graphs.

## Key findings

- Transforming a simple graph into an Eulerian graph by minimal switches is NP-hard.
- Any two Eulerian graphs can be connected via 2-switches with an Eulerian intermediate sequence.
- Finding the shortest 2-switch sequence between Eulerian graphs is NP-hard.

## Abstract

We show that the graph transformation problem of turning a simple graph into an Eulerian one by a minimum number of single edge switches is NP-hard. Further, we show that any simple Eulerian graph can be transformed into any other such graph by a sequence of 2-switches (i.e., exchange of two edge pairs), such that every intermediate graph is also Eulerian. However, finding the shortest such sequence also turns out to be an NP-hard problem.