The parameterized complexity of finding a 2-sphere in a simplicial complex

We consider the problem of finding a subcomplex \scrK \prime of a simplicial complex \scrK such that \scrK \prime is homeomorphic to the 2-dimensional sphere, \BbbS2. We study two variants of this problem. The first asks if there exists such a \scrK \prime with at most \scrK triangles, and we show that this variant is \sansW [\sansone ]-hard and, assuming the exponential time hypothesis, admits no no( \surdk)-time algorithm. We also give an algorithm that is tight with regard to this lower bound. The second problem is the dual of the first and asks if \scrK \prime can be found by removing at most k triangles from \scrK . This variant has an immediate \scrO (3kpoly(| \scrK | ))-time algorithm, and we show that it admits a polynomial kernelization to \scrO (k2) triangles, as well as a polynomial compression to a weighted version with bit-size \scrO (k log k).