Reconfiguration in bounded bandwidth and treedepth

TL;DR

This paper investigates the computational complexity of reconfiguration problems on graphs with bounded bandwidth and treedepth, showing PSPACE-completeness persists in the former and tractability in the latter.

Contribution

It demonstrates that certain reconfiguration problems remain PSPACE-complete on graphs of bounded bandwidth and become tractable on graphs of bounded treedepth, resolving an open question.

Findings

Reconfiguration problems are PSPACE-complete on bounded bandwidth graphs.

Reconfiguration problems are tractable on bounded treedepth graphs.

The results are tight, indicating limits of tractability.

Abstract

We show that several reconfiguration problems known to be PSPACE-complete remain so even when limited to graphs of bounded bandwidth. The essential step is noticing the similarity to very limited string rewriting systems, whose ability to directly simulate Turing Machines is classically known. This resolves a question posed open in [Bonsma P., 2012]. On the other hand, we show that a large class of reconfiguration problems becomes tractable on graphs of bounded treedepth, and that this result is in some sense tight.