  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.
