Rerouting Planar Curves and Disjoint Paths

TL;DR

This paper studies the complexity of transforming one set of disjoint paths into another within a graph, showing PSPACE-completeness in general but polynomial-time solutions for planar graphs with specific boundary conditions.

Contribution

It proves PSPACE-completeness of disjoint paths reconfiguration in general graphs and provides a polynomial-time algorithm for planar graphs with boundary-connected paths.

Findings

Disjoint Paths Reconfiguration is PSPACE-complete for general graphs.

Polynomial-time algorithm exists for planar graphs with boundary-connected paths.

Reconfiguration of disjoint s-t paths is polynomial in planar graphs but PSPACE-complete otherwise.

Abstract

In this paper, we consider a transformation of $k$ disjoint paths in a graph. For a graph and a pair of $k$ disjoint paths $\mathcal{P}$ and $\mathcal{Q}$ connecting the same set of terminal pairs, we aim to determine whether $\mathcal{P}$ can be transformed to $\mathcal{Q}$ by repeatedly replacing one path with another path so that the intermediates are also $k$ disjoint paths. The problem is called Disjoint Paths Reconfiguration. We first show that Disjoint Paths Reconfiguration is PSPACE-complete even when $k=2$. On the other hand, we prove that, when the graph is embedded on a plane and all paths in $\mathcal{P}$ and $\mathcal{Q}$ connect the boundaries of two faces, Disjoint Paths Reconfiguration can be solved in polynomial time. The algorithm is based on a topological characterization for rerouting curves on a plane using the algebraic intersection number. We also consider a transformation of disjoint $s$-$t$ paths as a variant. We show that the disjoint $s$-$t$ paths reconfiguration problem in planar graphs can be determined in polynomial time, while the problem is PSPACE-complete in general.