  We introduce the balanced crown decomposition that captures the structure
imposed on graphs by their connected induced subgraphs of a given size. Such
subgraphs are a popular modeling tool in various application areas, where the
non-local nature of the connectivity condition usually results in very
challenging algorithmic tasks. The balanced crown decomposition is a
combination of a crown decomposition and a balanced partition which makes it
applicable to graph editing as well as graph packing and partitioning problems.
We illustrate this by deriving improved kernelization and approximation
algorithms for a variety of such problems. In particular, through this
structure, we obtain the first constant-factor approximation for the Balanced
Connected Partition (BCP) problem, where the task is to partition a
vertex-weighted graph into $k$ connected components of approximately equal
weight. We derive a 3-approximation for the two most commonly used objectives
of maximizing the weight of the lightest component or minimizing the weight of
the heaviest component.
