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 𝒫 and 𝒬 connecting the same set of terminal pairs, we aim to determine whether 𝒫 can be transformed to 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 𝒫 and 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.
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