图着色问题的数据约简
Data Reduction for Graph Coloring Problems
摘要：
本文研究了图着色问题的核化复杂性。我们感兴趣的是如何以及多项式时间的数据减少可以证明缩小着色问题的实例，在所选择的参数。众所周知，决定3-可着色性已经是NP完全的，因此通过所请求的颜色数量进行参数化是没有成果的。相反，我们选择了Cai（DAM，2003）发起的一个研究线程，他研究了通过输入图到一个图类的修改距离参数化的着色问题，在这个图类上着色是多项式时间可解的;例如，通过使图弦所需的顶点删除数k参数化。我们得到各种上限和下限的内核的参数化的q-着色，补充蔡的研究的时间复杂度相对于这些参数。我们的研究结果表明，多项式核的存在性由顶点删除距离到一个图类F参数化的q-着色是强相关的函数f（q）的存在性，它的边界的顶点的数量，需要保持否答案的一个例子的q-列表着色F。
英文摘要：
This paper studies the kernelization complexity of graph coloring problems with respect to certain structural parameterizations of the input instances. We are interested in how well polynomial-time data reduction can provably shrink instances of coloring problems, in terms of the chosen parameter. It is well known that deciding 3-colorability is already NP-complete, hence parameterizing by the requested number of colors is not fruitful. Instead, we pick up on a research thread initiated by Cai (DAM, 2003) who studied coloring problems parameterized by the modification distance of the input graph to a graph class on which coloring is polynomial-time solvable; for example parameterizing by the number k of vertex-deletions needed to make the graph chordal. We obtain various upper and lower bounds for kernels of such parameterizations of q-Coloring, complementing Cai's study of the time complexity with respect to these parameters. Our results show that the existence of polynomial kernels for q-Coloring parameterized by the vertex-deletion distance to a graph class F is strongly related to the existence of a function f(q) which bounds the number of vertices which are needed to preserve the NO-answer to an instance of q-List-Coloring on F.
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