  We explore a reconfiguration version of the dominating set problem, where a
dominating set in a graph $G$ is a set $S$ of vertices such that each vertex is
either in $S$ or has a neighbour in $S$. In a reconfiguration problem, the goal
is to determine whether there exists a sequence of feasible solutions
connecting given feasible solutions $s$ and $t$ such that each pair of
consecutive solutions is adjacent according to a specified adjacency relation.
Two dominating sets are adjacent if one can be formed from the other by the
addition or deletion of a single vertex.
  For various values of $k$, we consider properties of $D_k(G)$, the graph
consisting of a vertex for each dominating set of size at most $k$ and edges
specified by the adjacency relation. Addressing an open question posed by Haas
and Seyffarth, we demonstrate that $D_{\Gamma(G)+1}(G)$ is not necessarily
connected, for $\Gamma(G)$ the maximum cardinality of a minimal dominating set
in $G$. The result holds even when graphs are constrained to be planar, of
bounded tree-width, or $b$-partite for $b \ge 3$. Moreover, we construct an
infinite family of graphs such that $D_{\gamma(G)+1}(G)$ has exponential
diameter, for $\gamma(G)$ the minimum size of a dominating set. On the positive
side, we show that $D_{n-m}(G)$ is connected and of linear diameter for any
graph $G$ on $n$ vertices having at least $m+1$ independent edges.
