  The concept of bounded expansion provides a robust way to capture sparse
graph classes with interesting algorithmic properties. Most notably, every
problem definable in first-order logic can be solved in linear time on bounded
expansion graph classes. First-order interpretations and transductions of
sparse graph classes lead to more general, dense graph classes that seem to
inherit many of the nice algorithmic properties of their sparse counterparts.
  In this work we introduce lacon- and shrub-decompositions and use them to
characterize transductions of bounded expansion graph classes and other graph
classes. If one can efficiently compute sparse shrub- or lacon-decompositions
of transductions of bounded expansion classes then one can solve every problem
definable in first-order logic in linear time on these classes.
