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Title: Reconfiguration of Connected Graph Partitions via Recombination
Motivated by applications in gerrymandering detection, we study a reconfiguration problem on connected partitions of a connected graph G. A partition of V(G) is connected if every part induces a connected subgraph. In many applications, it is desirable to obtain parts of roughly the same size, possibly with some slack s. A Balanced Connected k-Partition with slack s, denoted (k, s)-BCP, is a partition of V(G) into k nonempty subsets, of sizes n1,…,nk with |ni−n/k|≤s , each of which induces a connected subgraph (when s=0 , the k parts are perfectly balanced, and we call it k-BCP for short). A recombination is an operation that takes a (k, s)-BCP of a graph G and produces another by merging two adjacent subgraphs and repartitioning them. Given two k-BCPs, A and B, of G and a slack s≥0 , we wish to determine whether there exists a sequence of recombinations that transform A into B via (k, s)-BCPs. We obtain four results related to this problem: (1) When s is unbounded, the transformation is always possible using at most 6(k−1) recombinations. (2) If G is Hamiltonian, the transformation is possible using O(kn) recombinations for any s≥n/k , and (3) we provide negative instances for s≤n/(3k) . (4) We show that the problem is PSPACE-complete when k∈O(nε) and s∈O(n1−ε) , for any constant 0<ε≤1 , even for restricted settings such as when G is an edge-maximal planar graph or when k≥3 and G is planar.  more » « less
Award ID(s):
1800734
PAR ID:
10253553
Author(s) / Creator(s):
; ; ; ;
Date Published:
Journal Name:
Algorithms and Complexity (CIAC 2021)
Volume:
12701
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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