  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 $G$ 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. 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
the $i$-graph of $G$. We consider a class of graphs called "theta graphs": a
theta graph is the union of three internally disjoint nontrivial paths with the
same two distinct end vertices. We characterize theta graphs that are $i$-graph
realizable, showing that there are only finitely many that are not. We also
characterize those line graphs and claw-free graphs that are $i$-graphs, and
show that all $3$-connected cubic bipartite planar graphs are $i$-graphs.
