Name
Abstract | ||
|---|---|---|
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 \emph{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 \emph{Balanced Connected $k$-Partition with slack $s$}, denoted \emph{$(k,s)$-BCP}, is a partition of $V(G)$ into $k$ nonempty subsets, of sizes $n_1,\ldots , n_k$ with $|n_i-n/k|\leq s$, each of which induces a connected subgraph (when $s=0$, the $k$ parts are perfectly balanced, and we call it \emph{$k$-BCP} for short). A \emph{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 $s\geq 0$, 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 $s \ge n/k$, and (3) we provide negative instances for $s \leq n/(3k)$. (4) We show that the problem is PSPACE-complete when $k \in O(n^{\varepsilon})$ and $s \in O(n^{1-\varepsilon})$, for any constant $0 < \varepsilon \le 1$, even for restricted settings such as when $G$ is an edge-maximal planar graph or when $k=3$ and $G$ is planar. |
Year | DOI | Venue |
|---|---|---|
2021 | 10.1007/978-3-030-75242-2_4 | CIAC |
DocType | Citations | PageRank |
Conference | 0 | 0.34 |
References | Authors | |
0 | 5 | |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Hugo A. Akitaya | 1 | 11 | 9.76 |
| Matias Korman | 2 | 178 | 37.28 |
| Oliver Korten | 3 | 0 | 1.35 |
| Diane L. Souvaine | 4 | 480 | 77.99 |
| Csaba D. Toth | 5 | 91 | 13.63 |