Complement of automata

Question

I know that in order for the complement of the automaton to work, it needs to be deterministic and complete, and if it is not deterministic we can always apply the power set construction, and if is incomplete we can include a trash state. But ,what if I have a deterministic but incomplete automaton. I tried to construct some examples, but in all my examples the complement of the automata actually worked. Is there a few state example of an automaton which is deterministic but incomplete, and whose complement doesn't work? When I say doesn't work, I mean it gives wrong result.

Answer 1

Consider the language $L = (ab)^*$. An incomplete automaton for $L$ is given by $1 \xrightarrow{a} 2$ and $2 \xrightarrow{b} 1$, where $1$ is the initial state and unique final state. Now if you take $2$ as the final state, you accept the language $(ab)^*a$, which is not the complement of $L$.