At least one person was killed in clashes between security forces and supporters of ousted President Mohamed Morsi in Nasser, a medical official told Ahram Online.
Egyptian daily al-Masry al-Youm reported the person was killed trying to break into a polling station.
In the Nile Delta, al-Ahram Arabic reported five bombs were defused near several polling places.
Fighting also was reported between pro-Morsi and government forces throughout the country.
The Supreme Electoral Committee said it had opened extra polling stations because of the high turnout on the constitution, Ahram Online reported. The commission said it also has received complaints that the voting process was slow in some polling venues, so extra judges and civil servants were sent to help.
An electoral monitoring outpost received reports that supervising judges at some stations were urging voters to vote against the charter, Sherif Badr, head of the Egyptian Cabinet's Information and Decision Center, told MENA. The center said it also received reports of voters in poor villages reportedly being offered blankets in exchange for their ID cards to bar them from voting.
The 2012 constitution was amended as part of the July 3 road map, which included the ouster of President Mohamed Morsi, amending the Islamist drafted constitution -- which was heavily criticized for not being representative -- and parliamentary and presidential elections after the vote on the revised constitution.
In the run-up to Tuesday's vote, Egypt's interior minister warned any disruption of the vote on the country's revised Constitution would be crushed. Hundreds of thousands of well armed Egyptian soldiers and police guarded 35,000 polling stations across the country in a referendum on the draft charter the country's top military officer, Gen. Abdel Fattah al-Sisi, said would likely be a prelude to his bid for the presidency.
Security forces in Giza arrested al-Jazeera reporters covering the voting, seizing their cameras and other equipment, al-Ahram's Arabic website reported. Since Morsi's ouster, officials have cracked down on al-Jazeera, accusing it of biased reporting favoring the Muslim Brotherhood.
Al-Jazeera's offices in Egypt have been shuttered since Morsi's ouster and at least five reporters remain jailed.
Before voting began, Ahram Online reported an explosion at a courthouse in the Imbaba neighborhood of northern Giza. No one was reported injured but the courthouse's facade was destroyed.
Since Morsi's ouster in July, Egypt has experienced a number of bomb attacks. Authorities have said they linked the violence to the now-outlawed Muslim Brotherhood, of which Morsi is a member. The brotherhood has denied any tie to the violence.
The military-led interim regime has promised a return to civilian rule after the charter is approved and presidential and parliamentary elections are held.
The Muslim Brotherhood has called Tuesday's vote an attempt to legitimize the coup that ousted Morsi, Egypt's first freely elected president. Groups aligned with the Brotherhood have called for a voter boycott of the polls and anti-referendum demonstrations.
Suez Canal University political science Professor Gamal Zahran told Egyptian state newspaper al-Ahram he believes a strong yes vote for the revised Constitution would open the door to a Sisi presidential run.
"I think this week's vote will be as much about the popularity of Sisi as an endorsement of the new Constitution," he said.
Ziad al-Oleimi, a member of the youth coalition that forced President Hosni Mubarak from office in 2011, said he was unhappy about the prospect of a Sisi presidency.
He said Sisi -- who is Egypt's defense minister, armed forces commander in chief and first deputy prime minister -- would "return Egypt to military rule."
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