# Reconfiguring Undirected Paths

## TL;DR

This paper studies the problem of reconfiguring fixed-length paths in undirected graphs through edge additions and removals, providing efficient algorithms for trees and fixed-parameter cases, but proving PSPACE-completeness in certain graph classes.

## Contribution

It introduces algorithms for reconfiguring paths in trees and fixed-parameter cases, and establishes computational hardness in graphs of bounded bandwidth.

## Key findings

- Linear-time solution for trees
- Fixed-parameter tractability for certain parameters
- PSPACE-completeness in graphs of bounded bandwidth

## Abstract

We consider problems in which a simple path of fixed length, in an undirected graph, is to be shifted from a start position to a goal position by moves that add an edge to either end of the path and remove an edge from the other end. We show that this problem may be solved in linear time in trees, and is fixed-parameter tractable when parameterized either by the cyclomatic number of the input graph or by the length of the path. However, it is PSPACE-complete for paths of unbounded length in graphs of bounded bandwidth.