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. It is known that the problem admits an algorithm with running time f(c).n^{O(c)}, where n is the size of the graph G and c is the size of the two-dimensional complex C. In other words, that algorithm is polynomial when C is fixed, but the degree of the polynomial depends on C. We prove that the problem is fixed-parameter tractable in the size of the two-dimensional complex, by providing a deterministic f(c).n^3-time algorithm. We also provide a randomized algorithm with expected running time 2^{c^{O(1)}}.n^{O(1)}. 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; the only elaborated tool that we use is an algorithm to compute grid minors.

中文翻译：

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一种将图嵌入二维单纯复形的 FPT 算法

我们考虑图 G 到二维单纯复形 C 的可嵌入性问题：给定 G 和 C，决定 G 是否承认拓扑嵌入到 C 中。 该问题是 NP 难的，即使在 C 同构的受限情况下一个表面。已知该问题承认算法的运行时间为 f(c).n^{O(c)}，其中 n 是图 G 的大小，c 是二维复数 C 的大小。换句话说，当 C 固定时，该算法是多项式的，但多项式的次数取决于 C。我们通过提供确定性 f(c) 证明了该问题在二维复数的大小上是固定参数易处理的。 n^3 次算法。我们还提供了一个预期运行时间为 2^{c^{O(1)}}.n^{O(1)} 的随机算法。我们的方法是通过不相关的顶点方法减少 G 有界分支宽度的情况，并应用动态规划。我们不依赖现有线性时间算法的任何组件在固定表面上嵌入图；我们使用的唯一精心设计的工具是计算网格次要的算法。