The maximum common subgraph problem is an NP-hard problem which is very difficult to solve with exact approaches. To speedup the solution process, we may decompose it into independent sub-problems which are solved in parallel. We describe a new decomposition method which exploits the structure of the problem to decompose it. We compare this structural decomposition with domain-based decompositions, which basically split variable domains. Experimental results show us that the structural decomposition leads to better speedups on two classes of instances, and to worse speedups on one class of instances.