BAGHDAD, Iraq  — A string of attacks killed 19 people, including two relatives of a senior Kurdish official, and Vice President Dick Cheney made a surprise visit Sunday in which he suggested that Iraq's recent elections were a major step toward withdrawing U.S. troops.

Cheney's visit, under heavy security, was so secret that even Iraq's prime minister said he was surprised when he showed up for what he believed was a meeting with the U.S. ambassador only to see Cheney waiting to greet him.

Cheney toured the country the same day President Bush scheduled a prime-time address on Iraq.

"The participation levels all across the country were remarkable," Cheney told reporters after an hourlong briefing from the war's top military commanders. "And that's exactly what need to happen as you build a political structure in a self-governing Iraq that can unify the various segments of the population and ultimately take over responsibility for their own security."

The vice president visited with Iraq's leaders and military commanders in the Green Zone, saw an Iraqi troop training demonstration at Taji air base, lunched with soldiers who provided security for Thursday's election and gave a speech to troops.

The string of attacks late Saturday and early Sunday pierced three days of relative calm that followed the country's first election for a full-term parliament.

The violence, including two suicide bombings, came after authorities eased stringent security measures put in place for the Dec. 15 parliamentary election and traffic returned to normal on the first full working day since the vote.

A ban on vehicles was lifted and the country's borders reopened Saturday, although the frontier with Syria remained closed. Authorities said it would reopen in a few days and did not give a reason for the delay.

In the northern city of Kirkuk, two relatives of an official of the Patriotic Union of Kurdistan, one of the two main Kurdish parties, were shot late Saturday as they walked near their house, police said. They were identified as Dhiab Hamad al-Hamdani and his son — the uncle and nephew of party official Khodr Hassan al-Hamdani. The PUK is led by President Jalal Talabani.

In Baghdad on Sunday, a roadside bomb killed three police officers and wounded two. A similar attack Saturday night killed one policeman and wounded two in the northern town of Tuz, 68 miles south of Kirkuk, police said.

Unidentified gunmen killed a police officer and an Interior Ministry employee in separate attacks. Both were driving to work in western Baghdad when they were attacked. Four police officers were seriously injured when their squad car was sprayed with gunfire, and a tea seller was shot and killed in the same area.

A police captain and his driver were shot and killed in south Baghdad while two people, including an Interior Ministry driver, were killed in Baghdad's Shiite Sadr City slum.

A suicide bomber killed a police officer and injured two when he blew up a bomb in a minivan at a checkpoint along a Baghdad highway near the Interior Ministry.

A roadside bomb killed at least one woman and injured 11 in the northern Shiite neighborhood of Kazimiyah, police said.

Police also said a suicide bomber was killed in Amiriyah, about 25 miles west of Baghdad, when his explosives-laden belt prematurely detonated.

On Sunday, police found the body of a former Iraqi Army officer at a fuel station in central Baghdad. Abbas Abdullah Fadhl had been shot to death in his car, they said. Another unidentified man was found shot dead in east Baghdad.

Millions of Iraqis voted Thursday to choose a four-year parliament in an election that passed peacefully around the country. Although no official figures have been released, authorities estimate just under 70 percent of Iraq's 15 million registered voters cast ballots.

The big turnout — particularly among the disaffected Sunni Arab minority that boycotted January's election of a temporary legislature — have boosted hopes that increasing political participation may undermine the insurgency and allow U.S. troops to begin pulling out next year.

Shiites account for about 60 percent of the country's estimated 27 million people, compared with 20 percent for Sunni Arabs. Both Shiite and Sunni political leaders have said they likely will have to form a coalition government together.

Shiite Arabs and Kurds, two groups oppressed under the Sunni Arab-dominated regime of Saddam Hussein, allied to form the interim government that has ruled since last spring.