  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.
