# Shortest Reconfiguration of Perfect Matchings via Alternating Cycles

## TL;DR

This paper investigates the shortest reconfiguration sequences between perfect matchings via alternating cycles, revealing computational complexity results and polynomial-time solutions for specific graph classes.

## Contribution

It establishes NP-hardness for the problem in general and planar or bipartite graphs, and provides a polynomial-time algorithm for outerplanar graphs.

## Key findings

- NP-hardness in planar and bipartite graphs
- Polynomial-time algorithm for outerplanar graphs
- Connection to perfect matching polytopes

## Abstract

Motivated by adjacency in perfect matching polytopes, we study the shortest reconfiguration problem of perfect matchings via alternating cycles. Namely, we want to find a shortest sequence of perfect matchings which transforms one given perfect matching to another given perfect matching such that the symmetric difference of each pair of consecutive perfect matchings is a single cycle. The problem is equivalent to the combinatorial shortest path problem in perfect matching polytopes. We prove that the problem is NP-hard even when a given graph is planar or bipartite, but it can be solved in polynomial time when the graph is outerplanar.