We develop a framework for applying treewidth-based dynamic programming on graphs with “hybrid structure”, i.e., with parts that may not have small treewidth but instead possess other structural properties. Informally, this is achieved by defining a refinement of treewidth which only considers parts of the graph that do not belong to a pre-specified tractable graph class. Our approach allows us to not only generalize existing fixed-parameter algorithms exploiting treewidth, but also fixed-parameter algorithms which use the size of a modulator as their parameter. As the flagship application of our framework, we obtain a parameter that combines treewidth and rank-width to obtain fixed-parameter algorithms for Chromatic Number, Hamiltonian Cycle, and Max-Cut.

我们开发了一个框架，用于在具有“混合结构”的图上应用基于树宽的动态编程，即，零件的树宽可能不小，而是具有其他结构属性。非正式地，这是通过定义树宽的细化来实现的，该细化仅考虑图的不属于预定可处理图类的部分。我们的方法不仅使我们能够推广利用树宽的现有固定参数算法，而且能够将使用调制器大小作为参数的固定参数算法进行泛化。作为我们框架的旗舰应用程序，我们获得了一个结合树宽和秩宽的参数，以获得色数，哈密​​顿循环和最大剪切的固定参数算法。