  Vertex deletion problems ask whether it is possible to delete at most $k$
vertices from a graph so that the resulting graph belongs to a specified graph
class. Over the past years, the parameterized complexity of vertex deletion to
a plethora of graph classes has been systematically researched. Here we present
the first single-exponential fixed-parameter tractable algorithm for vertex
deletion to distance-hereditary graphs, a well-studied graph class which is
particularly important in the context of vertex deletion due to its connection
to the graph parameter rank-width. We complement our result with matching
asymptotic lower bounds based on the exponential time hypothesis. As an
application of our algorithm, we show that a vertex deletion set to
distance-hereditary graphs can be used as a parameter which allows
single-exponential fixed-parameter tractable algorithms for classical NP-hard
problems.
