  In this paper, we study the graphical structure of elementary trapping sets
(ETS) of variable-regular low-density parity-check (LDPC) codes. ETSs are known
to be the main cause of error floor in LDPC coding schemes. For the set of LDPC
codes with a given variable node degree $d_l$ and girth $g$, we identify all
the non-isomorphic structures of an arbitrary class of $(a,b)$ ETSs, where $a$
is the number of variable nodes and $b$ is the number of odd-degree check nodes
in the induced subgraph of the ETS. Our study leads to a simple
characterization of dominant classes of ETSs (those with relatively small
values of $a$ and $b$) based on short cycles in the Tanner graph of the code.
For such classes of ETSs, we prove that any set ${\cal S}$ in the class is a
layered superset (LSS) of a short cycle, where the term "layered" is used to
indicate that there is a nested sequence of ETSs that starts from the cycle and
grows, one variable node at a time, to generate ${\cal S}$. This
characterization corresponds to a simple search algorithm that starts from the
short cycles of the graph and finds all the ETSs with LSS property in a
guaranteed fashion. Specific results on the structure of ETSs are presented for
$d_l = 3, 4, 5, 6$, $g = 6, 8$ and $a, b \leq 10$ in this paper. The results of
this paper can be used for the error floor analysis and for the design of LDPC
codes with low error floors.
