Reconfiguration of connected graph partitions via recombination

Motivated by applications in gerrymandering detection, we study a reconfiguration problem on connected partitions of graph G. A partition V(G) is if every part induces subgraph. In many applications, it desirable to obtain parts roughly the same size, possibly with some slack s. Balanced Connected k-Partition s, denoted (k,s)-BCP, into k nonempty subsets, sizes n1,…,nk |ni−n/k|≤s, each which subgraph (when s=0, are perfectly balanced, and call k-BCP for short). recombination an operation that takes (k,s)-BCP G produces another merging two adjacent subgraphs repartitioning them. Given k-BCPs, B, s≥0, wish determine whether there exists sequence recombinations transform B via (k,s)-BCPs. We four results related this problem: (1) When s unbounded, transformation always possible using at most 6(k−1) recombinations. (2) If Hamiltonian, O(kn) any s≥n/k, (3) exist negative instances s≤n/(3k), (4) show determining connects PSPACE-complete when k∈O(nε) s∈O(n1−ε), constant 0<ε≤1. This statement holds even restricted settings such as edge-maximal planar or k≥3 planar.