Reconfiguration of Connected Graph Partitions
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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 NP-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 length is known to exist). We also present efficient algorithms for computing a sequence of 1-switches that takes 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.
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