# Reconfiguration of Connected Graph Partitions

## TL;DR

This paper investigates the reconfiguration of connected graph partitions into districts, providing combinatorial characterizations, complexity results, and algorithms for transforming one partition into another via minimal vertex switches.

## Contribution

It offers a combinatorial criterion for the connectivity of district maps, proves NP-completeness of reconfiguration problems, and develops efficient algorithms for connected cases.

## Key findings

- Connectedness can be tested efficiently.
- Deciding the existence of a reconfiguration sequence is NP-complete.
- Algorithms perform near-optimal reconfiguration with minimal switches.

## Abstract

Motivated by recent computational models for redistricting and detection of gerrymandering, we study the following problem on graph partitions. Given a graph $G$ and an integer $k\geq 1$, a $k$-district map of $G$ is a partition of $V(G)$ into $k$ nonempty subsets, called districts, each of which induces a connected subgraph of $G$. A switch is an operation that modifies a $k$-district map by reassigning a subset of vertices from one district to an adjacent district; a 1-switch is a switch that moves a single vertex. We study the connectivity of the configuration space of all $k$-district maps of a graph $G$ under 1-switch operations. We give a combinatorial characterization for the connectedness of this space that can be tested efficiently. We prove that it is NP-complete to decide whether there exists a sequence of 1-switches that takes a given $k$-district map into another; and NP-hard to find the shortest such sequence (even if a sequence of polynomial length is known to exist). We also present efficient algorithms for computing a sequence of 1-switches that takes a given $k$-district map into another when the space is connected, and show that these algorithms perform a worst-case optimal number of switches up to constant factors.