DFA minimization
Every regular language has exactly one smallest DFA. Draw any DFA for the language and watch it shrink: the groups of states split step by step until only truly different states remain.
This DFA is correct but wasteful: q3 does the same job as q0, q4 the same as q1, and q5 cannot be reached at all. States that still count as the same share a colour; watch the colours split until only real differences remain, then the groups merge into the three states the language needs.
Remove states that can never be reached
Anything not reachable from q0 has no effect on the language, so it goes first.
Split the states into accepting and non-accepting
These two groups are certainly different: the empty string already tells them apart.
Keep splitting groups whose members disagree
Two states stay together only if, for every symbol, they lead into the same group. Whenever they do not, the group is split. This repeats until nothing changes.
Merge each remaining group into one state
States still together cannot be told apart by any input, so one state can do the work of all of them.
Does the DFA have to be complete?
Yes — every state needs a transition for every symbol, which is what the simulator requires of a DFA anyway. It tells you which states are missing one.
What if my DFA is already minimal?
You are told so, and nothing new is opened. That is a quick way to check a hand-built answer.
Can I minimize an NFA?
Convert it to a DFA first, then minimize the result. Minimal NFAs are a much harder problem and are not unique.