# Independent Set Reconfiguration Parameterized by Modular-Width

## TL;DR

This paper proves that the Independent Set Reconfiguration problem is fixed-parameter tractable when parameterized by modular-width, resolving an open problem and expanding understanding of reconfiguration complexity.

## Contribution

It demonstrates fixed-parameter tractability of the problem under modular-width for all studied reconfiguration rules, including resolving an open problem for TAR.

## Key findings

- Problem is FPT under modular-width for TAR, TJ, TS rules.
- Resolves an open problem for TAR reconfiguration.
- Shows modular-width parameterization differs from bandwidth.

## Abstract

Independent Set Reconfiguration is one of the most well-studied problems in the setting of combinatorial reconfiguration. It is known that the problem is PSPACE-complete even for graphs of bounded bandwidth. This fact rules out the tractability of parameterizations by most well-studied structural parameters as most of them generalize bandwidth. In this paper, we study the parameterization by modular-width, which is not comparable with bandwidth. We show that the problem parameterized by modular-width is fixed-parameter tractable under all previously studied rules TAR, TJ, and TS. The result under TAR resolves an open problem posed by Bonsma [WG 2014, JGT 2016].