The class of graph deletion problems has been extensively studied in theoretical computer science, particularly in the field of parameterized complexity. Recently, a new notion of graph deletion problems was introduced, called deletion to scattered graph classes, where after deletion, each connected component of the graph should belong to at least one of the given graph classes. While fixed-parameter algorithms were given for a wide variety of problems, little progress has been made on the kernelization complexity of any of them. In this paper, we present the first non-trivial polynomial kernel for one such deletion problem, where, after deletion, each connected component should be a clique or a tree - that is, as densest as possible or as sparsest as possible (while being connected). We develop a kernel consisting of O(k^5) vertices for this problem.

翻译：图删除问题类在理论计算机科学中得到了广泛研究，特别是在参数化复杂性领域。近年来，引入了一种新的图删除问题概念，称为“散乱图类删除”，即删除后图的每个连通分量应至少属于给定的图类之一。尽管针对多种问题设计了固定参数算法，但其中任意问题的核化复杂性均进展甚微。本文针对一个具体的删除问题首次给出了非平凡的多项式核：要求删除后每个连通分量要么是团（尽可能密集）要么是树（尽可能稀疏，同时保持连通）。我们为该问题开发了一个包含O(k^5)个顶点的核。