  We present a dynamic data structure for representing a graph $G$ with
tree-depth at most $D$. Tree-depth is an important graph parameter which arose
in the study of sparse graph classes.
  The structure allows addition and removal of edges and vertices such that the
resulting graph still has tree-depth at most $D$, in time bounds depending only
on $D$. A tree-depth decomposition of the graph is maintained explicitly.
  This makes the data structure useful for dynamization of static algorithms
for graphs with bounded tree-depth. As an example application, we give a
dynamic data structure for MSO-property testing, with time bounds for removal
depending only on $D$ and constant-time testing of the property, while the time
for the initialization and insertion also depends on the size of the formula
expressing the property.
