Reconfiguration of connected graph partitions

Motivated by recent computational models for redistricting and detection of gerrymandering, we study the following problem on graph partitions. Given a graph G and an integer k ≥ 1 , a k ‐ district map of G is a partition of V ( G )into k nonempty subsets, called districts , each of which induces a connected subgraph of G . A switch is an operation that modifies a k ‐district map by reassigning a subset of vertices from one district to an adjacent district; a 1‐switch is a switch that moves a single vertex. We study the connectivity of the configuration space of all k ‐district maps of a graph G under 1‐switch operations. We give a combinatorial characterization for the connectedness of this space that can be tested efficiently. We prove that it is PSPACE‐complete to decide whether there exists a sequence of 1‐switches that takes a given k ‐district map into another; and NP‐hard to find the shortest such sequence (even if a sequence of polynomial lengths is known to exist). We also present efficient algorithms for computing a sequence of 1‐switches that take a given k ‐district map into another when the space is connected, and show that these algorithms perform a worst‐case optimal number of switches up to constant factors.