Common subgraph isomorphism with K vertex
The fact that the Maximum Common Subgraph is NP-complete suggests that your "dumb" algorithm is about as good as it gets. You certainly cannot expect a polynomial time solution.

There are a few approaches you could try, depending on the exact context you are working in, the details of which are more than can reasonably be reproduced here, but a good starting point would be Abu-Khzam et al.'s paper "The Maximum Common Subgraph Problem: Faster Solutions via Vertex Cover" (In Proceedings of AICCSA 2007, pp. 367-373). This describes a variety of exact approaches which exploit different properties of the input.