Rerouting Planar Curves and Disjoint Paths

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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.