Hardness of finding domination number of an arbitrary graph It is known that finding a minimum dominating set and domination number of an arbitrary graph is NP-complete. However, if the domination number is known, is there any (efficient) algorithm to find a minimum dominating set?

If there would be such (polynomial) algorithm for a given $n$, there would be a polynomial algorithm for the original problem, since there is only a polynomial number of possible values for $n$, and you can try them all, since verification is polynomial.

PS. Write 'Complexity' instead of 'Hardness'.