Reconfiguration of connected graph partitions

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