  Suppose that we are given two dominating sets $D_s$ and $D_t$ of a graph $G$
whose cardinalities are at most a given threshold $k$. Then, we are asked
whether there exists a sequence of dominating sets of $G$ between $D_s$ and
$D_t$ such that each dominating set in the sequence is of cardinality at most
$k$ and can be obtained from the previous one by either adding or deleting
exactly one vertex. This problem is known to be PSPACE-complete in general. In
this paper, we study the complexity of this decision problem from the viewpoint
of graph classes. We first prove that the problem remains PSPACE-complete even
for planar graphs, bounded bandwidth graphs, split graphs, and bipartite
graphs. We then give a general scheme to construct linear-time algorithms and
show that the problem can be solved in linear time for cographs, trees, and
interval graphs. Furthermore, for these tractable cases, we can obtain a
desired sequence such that the number of additions and deletions is bounded by
$O(n)$, where $n$ is the number of vertices in the input graph.
