The Parameterized Complexity of Finding a 2-Sphere in a Simplicial Complex

We consider the problem of finding a subcomplex K' of a simplicial complex K such that K' is homeomorphic to the 2-dimensional sphere, S^2. We study two variants of this problem. The first asks if there exists such a K' with at most k triangles, and we show that this variant is W[1]-hard and, assuming ETH, admits no O(n^(o(sqrt(k)))) time algorithm. We also give an algorithm that is tight with regards to this lower bound. The second problem is the dual of the first, and asks if K' can be found by removing at most k triangles from K. This variant has an immediate O(3^k poly(|K|)) time algorithm, and we show that it admits a polynomial kernelization to O(k^2) triangles, as well as a polynomial compression to a weighted version with bit-size O(k log k).