Mohammad Arif Noori, spokesman for the provincial governor in Ghazni, said the Taliban fighters hit the first checkpoint of a district governor house's compound that left five Afghan security forces dead and 26 others injured, including the district governor.
Elsewhere, militants raided a government building in the eastern city of Jalalabad in Nangarhar province, prompting terrified employees to jump out of the windows, officials said.
He said the militants had seized control of the district.
According to the Afghan president's spokesman Shahussain Murtazawi, the two sides discussed the implementation of the Afghanistan-Pakistan Action Plan for Peace, the fight against terrorism, reducing violence and Afghan peace process.
Both attacks were claimed by IS.
At least 12 people were killed and 31 wounded when a suicide attacker blew himself up outside a government ministry in Kabul on Monday, officials said, as employees were leaving their offices early for Ramadan.
"A suicide bomber detonated his explosive vest at the entrance gate of the ministry. killing and injuring a number of the employees of the ministry", Stanikzai told AFP.
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ISIS has reportedly claimed responsibility for the attack, although there's no evidence of the terror group's involvement.
"The Government of the Islamic Republic of Afghanistan announces a ceasefire from the 27th of Ramadan until the fifth day of Eid-ul-Fitr following the historic ruling (Fatwa) of the Afghan Ulema", said Ghani. Pakistan has denied the accusations.
The Taliban on Saturday issued a statement saying it had ordered its fighters not to clash with Afghan security forces for the first three days of Eid.
President Ashraf Ghani on Thursday announced a ceasefire with the Taliban after a religious scholars meeting in Kabul last Monday where they issued a fatwa against the ongoing war in the country. They said they would continue attacking US-led North Atlantic Treaty Organisation troops.
Afghan security forces and the Taliban have vowed to retaliate if attacked during the ceasefire.
If it holds, analysts had expressed cautious optimism that the agreements could help build trust between the government and the Taliban and lay the groundwork for peace talks.