Applying for F1 for graduate school. Question about 30 days overstay on previous F1.

I graduated from an undergraduate program in May 2015 and was preparing to apply for my OPT. However, in July, my grandfather passed away and I was contemplating returning to India. My parents convinced me to stay on and complete an internship to gain experience. In mid-July, my grandmother was also hospitalized. In the meantime, I managed to find an internship but decided against staying as I wanted to spend time with my grandmother and family during this troubled time. But my parents were not at home and could not send me a return ticket until after the grace period had passed. I left the U.S. voluntarily 30 days after the grace period expired.

Now I am applying for a fresh F1 visa for graduate school. All other factors are satisfactorily covered - financial assistantship, choice of school and course. Will I be denied on the basis of a brief overstay? Also, on the DS-160, should I mention the overstay in the background security question that is asked towards the end of the form?

Note: I did not accrue unlawful presence and was not contacted by USCIS during those 30 days. I was not deported but left voluntarily.

The ‘unlawful presence’ statute is not invoked in the present case because you seem to have left the USA before that. You should be okay. The answer in DS160 is ‘no’ to the question if you ever overstayed your visa.