  In parameterized complexity, it is well-known that a parameterized problem is
fixed-parameter tractable if and only if it has a kernel - an instance
equivalent to the input instance, whose size is just a function of the
parameter. The size of the kernel can be exponential or worse, resulting in a
quest for fixed-parameter tractable problems with a polynomial-sized kernel.
The developments in machinery to show lower bounds for the sizes of the kernel
gave rise to the question of the asymptotically optimum size for the kernel of
fixed-parameter tractable problems. In this article, we survey a tool called
expansion lemma that helps in reducing the size of the kernel. Its early origin
is in the form of Crown Decomposition for obtaining linear kernel for the
Vertex Cover problem and the specific lemma was identified as the tool behind
an optimal kernel with O(k^2) vertices and edges for the UNDIRECTED FEEDBACK
VERTEX SET problem. Since then, several variations and extensions of the tool
have been discovered. We survey them along with their applications in this
article.
