Rerouting Planar Curves and Disjoint Paths

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 \(\mathsf{PSPACE}\) -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 \(\mathsf{PSPACE}\) -complete in general.