图嵌入二维单纯复形的FPT算法
An FPT algorithm for the embeddability of graphs into two-dimensional simplicial complexes
AI总结：
针对图嵌入二维单纯复形的NP难问题，提出基于无关顶点法和动态规划的FPT算法（O(2^{poly(c)}·n²)时间），可计算嵌入表示，归约多个已知问题，结合现有结果可优化至线性时间。
摘要：
我们考虑图G嵌入二维单纯复形C的可嵌入性问题：给定G和C，判断G是否允许拓扑嵌入到C中。该问题即使在C同胚于曲面的受限情况下也是NP难的。我们通过提供一个O(2^{poly(c)}·n²)时间的算法，证明了该问题在二维复形的大小上是固定参数可解的。如果G可嵌入到C中，我们可以在相同时间内计算出嵌入的表示。此外，我们表明若干已知问题可归约为此问题，例如交叉数问题和平面性数问题，以及在某些条件下的嵌入扩展问题。我们的方法是通过无关顶点法将问题归约到G具有有界分支宽度的情况，并应用动态规划。我们不依赖现有固定曲面上嵌入图的线性时间算法的任何组件，仅依赖图子式理论中的算法。然而，通过将我们的结果与曲面上嵌入图的线性时间算法以及无关顶点法的最新结果相结合，我们可以在f(c)·O(n)时间内判断G是否可嵌入到C中，其中f是某个函数。
英文摘要：
We consider the embeddability problem of a graph G into a two-dimensional simplicial complex C: Given G and C, decide whether G admits a topological embedding into C. The problem is NP-hard, even in the restricted case where C is homeomorphic to a surface. We prove that the problem is fixed-parameter tractable in the size of the two-dimensional complex, by providing an O(2^{poly(c)}.n^2)-time algorithm. If G embeds into C, we can compute a representation of an embedding in the same amount of time. Moreover, we show that several known problems reduce to this one, such as the crossing number and the planarity number problems, and, under some conditions, the embedding extension problem. Our approach is to reduce to the case where G has bounded branchwidth via an irrelevant vertex method, and to apply dynamic programming. We do not rely on any component of the existing linear-time algorithms for embedding graphs on a fixed surface, but only on algorithms from graph minor theory. However, by combining our results with a linear-time algorithm for embedding graphs on surfaces and with a very recent result for the irrelevant vertex method, we can decide whether G embeds into C in f(c).O(n) time, for some function f.
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