There is no such thing as claiming a dependent for half the year. You either claim a dependent for the whole tax year or not at all. Are you asking about tax year 2020 or the upcoming 2021 return you will file in 2022?
The year of college graduation is tricky----if he was still a full-time student for at least five months of 2021 and he was under the age of 24, you can still claim him as a dependent. You will receive the $500 credit for other dependents and possibly an education credit.
Or....since he began working full time in 2021 .... you can agree to have him file a 2021 return that says he cannot be claimed as a dependent for 2021 if he earns over $4350 in 2021.
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