There were no immediate reports of any casualties from the blasts, which were condemned by both Fatah and its Islamist rival Hamas, which dominates the Gaza Strip.
The stage for commemorations of next week's 10th anniversary of the death of iconic Palestinian leader and Fatah founder Yasser Arafat was also targeted in the blasts which struck in the space of a few hours before 6:30 am (0430 GMT).
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It is the first time in years that a public commemoration of Arafat's death has been planned in Gaza.
Abbas loyalists were ousted from the territory in 2007 and rival administrations ruled in the West Bank and Gaza.
But earlier this year Hamas and Fatah struck a reconciliation agreement leading to the formation of a government of national unity.