Fixed structure complexity

We consider a non-standard parametrization, where, for problems consisting of a combinatorial structure and a number, we parameterize by the combinatorial structure, rather than by the number. For example, in the Short- Nondeterministic-Halt problem, which is to determine if a nondeterministic machine M accepts the empty string in t steps, we parameterize by |M|, rather than t. We call such parametrization fixed structure parametrization. Fixed structure parametrization not only provides a new set of parameterized problems, but also results in problems that do not seem to fall within the classical parameterized complexity classes. In this paper we take the first steps in understanding these problems. We define fixed structure analogues of various classical problems, including graph problems, and provide complexity, hardness and equivalence results.