  In this paper, we propose a characterization for non-elementary trapping sets
(NETSs) of low-density parity-check (LDPC) codes. The characterization is based
on viewing a NETS as a hierarchy of embedded graphs starting from an ETS. The
characterization corresponds to an efficient search algorithm that under
certain conditions is exhaustive. As an application of the proposed
characterization/search, we obtain lower and upper bounds on the stopping
distance $s_{min}$ of LDPC codes.
  We examine a large number of regular and irregular LDPC codes, and
demonstrate the efficiency and versatility of our technique in finding lower
and upper bounds on, and in many cases the exact value of, $s_{min}$. Finding
$s_{min}$, or establishing search-based lower or upper bounds, for many of the
examined codes are out of the reach of any existing algorithm.
