Israeli forces killed two men in raids in the Gaza Strip, one of whom was driving pregnant neighbour to hospital.
Israel had suspended attack operations during the elections
The Israeli army says the man sped towards a platoon in Beit Lahiya as they were hunting for wanted militants.
Another man was killed as troops raided the Bureij refugee camp. The Israeli army said he was armed.
The violence follows renewed efforts to resolve the Middle East conflict after the election of Palestinian leader Mahmoud Abbas, considered a moderate.
Mr Abbas said on Thursday that he was eager to restart talks on the international peace plan known as the road map which has been stalled since its launch in 2003.
The road map requires that Israel freezes settlement activity in the occupied West Bank and Gaza, while Palestinians must dismantle the militant groups.
"We are ready to implement our commitments. We hope the Israeli side will do the same," he told a visiting group of Christian leaders in Ramallah in the West Bank.
In another incident in Gaza, the Israeli air force fired three missiles at a Palestinian car parked outside a municipal building close to the Kfar Darom settlement. No one was injured but several nearby houses were damaged, Palestinian sources said. The Israeli army said the car was suspicious and may have contained explosives.
Correspondents say Mr Abbas needs the backing of militant groups Hamas and Islamic Jihad, who have been cautious about calls for a cease-fire with Israel.
European Union foreign policy chief Javier Solana met officials from the Israeli government on Thursday after spending the last few days in the West Bank and the Gaza Strip.
The EU's Solana is in the region to see what can be done for peace
Mr Solana is here to see for himself if this is really a new beginning in the Israeli-Palestinian conflict and to see if there is anything that Europe can do to help, says the BBC's James Reynolds in Jerusalem.
In particular, he is likely ask the Israeli government to support Mr Abbas and to give his administration a chance to succeed.
Israeli forces have occupied the Gaza Strip and West Bank - home to nearly 4m Palestinians - since 1967.
Israel is planning to unilaterally withdraw from Gaza this year but will keep control of its borders, coastline and airspace after the pullout.