Structural graph parameters play an important role in parameterized complexity, including in kernelization. Notably, vertex cover, neighborhood diversity, twin-cover, and modular-width have been studied extensively in the last few years. However, there are many fundamental problems whose preprocessing complexity is not fully understood under these parameters. Indeed, the existence of polynomial kernels or polynomial Turing kernels for famous problems such as Clique, Chromatic Number, and Steiner Tree has only been established for a subset of structural parameters. In this work, we use several techniques to obtain a complete preprocessing complexity landscape for over a dozen of fundamental algorithmic problems.

翻译：结构图参数在参数化复杂性中扮演重要角色，尤其在核化领域。近年来，顶点覆盖、邻域多样性、孪生覆盖和模宽等参数得到了广泛研究。然而，许多基本问题的预处理复杂性在这些参数下尚未完全明确。事实上，对于Clique、Chromatic Number和Steiner Tree等著名问题，多项式核或多项式图灵核的存在性仅针对部分结构参数得到确认。在本工作中，我们运用多种技术，为十余个基本算法问题描绘了完整的预处理复杂性图景。