Two Kabul University professors were among those arrested by the NDS in an operation against an ISIS cell.
The son of deceased al-Qaeda leader Osama bin Laden died sometime within the past two years, confirmed US officials.
The blast killed at least 34 people, mostly women and children, and wounded at least 17, officials said.
Many of the estimated 400 Turkmen militants who fought for ISIS in Syria and Iraq and survived are now in parts of Afghanistan that border Turkmenistan.
At least 20 people were killed and 50 others wounded in an attack targeting the Kabul office of President Ashraf Ghani's running mate, Amrullah Saleh.
Taliban attacks on schools and police alike show the group's lack of direction and low regard for human life.
Taliban attacks earlier this month killed and wounded scores of children, causing trauma and suffering that could last a lifetime, say observers.
The blast comes less than a week after 50 other children were wounded in another wayward Taliban attack in Kabul that damaged five schools.
Taliban militants killed dozens and wounded more than 100 -- including 50 children -- in three separate attacks across Afghanistan in recent days.
ISIS fighters who waged a bloody campaign in Iraq and Syria are heading to Afghanistan to continue their violent insurgency and help plot attacks against the United States.