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.

翻译：在参数化复杂性中，众所周知，一个参数化问题是固定参数可解的，当且仅当其有一个核——一个等价于输入实例且大小仅为参数函数的实例。核的规模可能呈指数级或更差，从而引发了对具有多项式规模核的固定参数可解问题的探索。核规模下界证明机制的发展引出了固定参数可解问题核的渐近最优规模问题。本文综述了一种名为“扩展引理”的工具，它有助于缩减核的规模。该工具早期以皇冠分解的形式出现，用于获得顶点覆盖问题的线性核，而后被确认为无向反馈顶点集问题最优核（含O(k^2)个顶点和边）背后的关键引理。自此，该工具的各种变体和扩展被相继发现。本文系统回顾了这些变体及其应用。