🤖 AI Summary
This paper addresses the topological embedding problem: deciding whether a graph (G) admits a continuous injective map into a 2-dimensional simplicial complex (C). Even when (C) is a surface, the problem is NP-hard. We establish, for the first time, that it is fixed-parameter tractable (FPT) with respect to the size (c) of (C), and present the first FPT algorithm: leveraging graph minor theory, we construct an irrelevant vertex framework, then integrate branchwidth reduction with dynamic programming to achieve (O(2^{ ext{poly}(c)} cdot n^2)) time for both decision and construction; further refinement yields an (f(c) cdot O(n)) linear-time algorithm. Our framework unifies classical problems—including crossing number and planarity testing—as special cases, and supports explicit output of the embedding.
📝 Abstract
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.