  The independent domination number $i(G)$ of a graph $G$ is the minimum
cardinality of a maximal independent set of $G$, also called an $i(G)$-set. The
$i$-graph of $G$, denoted $\mathcal{I}(G)$, is the graph whose vertices
correspond to the $i(G)$-sets, and where two $i(G)$-sets are adjacent if and
only if they differ by two adjacent vertices. We show that not all graphs are
$i$-graph realizable, that is, given a target graph $H$, there does not
necessarily exist a source graph $G$ such that $H$ is isomorphic to
$\mathcal{I}(G)$. Examples of such graphs include $K_{4}-e$ and $K_{2,3}$. We
build a series of tools to show that known $i$-graphs can be used to construct
new $i$-graphs and apply these results to build other classes of $i$-graphs,
such as block graphs, hypercubes, forests, cacti, and unicyclic graphs.
