Calculating the domination number is an NP-Hard problem. Does it remain NP-Hard if we restrict it to non-bipartite graphs?
The general case of the domination number problem reduces to the non-bipartite case. Given a (possibly bipartite graph) $G$, create new graph $G'$ consisting of an copy of $G$ and a disjoint copy of $K_3$. The domination number of the new (non-bipartite) graph $G'$ is exactly one more than that of the original graph $G$. So computing the domination number of the new graph $G'$ is exactly as hard as computing that of the original graph $G$.