By KEVIN FREKINGAssociated Press WASHINGTON (AP) – Democrats are hoping that late campaign appearances by former Rep. Gabrielle Giffords will help push her hand-picked successor to victory in a special congressional election in Arizona. Republicans, trying to make the southern Arizona House race a referendum on President Barack Obama and his handling of the economy, are running a former Marine who narrowly lost to Giffords two years ago. The Arizona race is just one of the election battles taking place across the nation Tuesday. Voters in Virginia, Maine, Nevada, North Dakota and South Carolina are taking part in primary elections. Giffords, 42, resigned in January to focus on her recovery from a gunshot wound to her head during a gunman's shooting spree a year earlier. Six people died and 13 were wounded at a constituent event she was hosting outside a Tucson supermarket. She has made few public appearances since the shooting but has returned to Tucson in recent days to help the former director of her district office, Ron Barber, in his race to succeed her in the House. His Republican opponent is Jesse Kelly. Giffords attended a concert Saturday night and listened as her husband, astronaut Mark Kelly, spoke on her behalf in praising Barber, who was wounded in the shooting. The couple met Sunday with volunteers to thank them for their work. “Some of those people used to be her supporters,” Kelly said after the meeting. “But now Ron has his own team that's energized to make sure he gets across the finish line on Tuesday, and Gabby is very excited about that.” The Republican candidate, Jesse Kelly – he is not related to Giffords' husband – continued to make the case in the election's final hours that Barber and Obama are out of touch with people in Arizona's 8th Congressional District. Republicans have a 26,000-person edge over Democrats in voter registrations. “It's time to put a stop to the Barber-Obama team,” Kelly's campaign said in its final ad. Outside groups have spent more than $2 million on the Arizona race. Barber, 66, had a sizable fundraising lead in late May, but spending from conservative groups helped reduce it. The Arizona 8th is a rare swing district that is competitive virtually every election. Giffords defeated Kelly by about 4,000 votes in 2010 when the election focused on immigration and when tea partiers rallied to the tough-talking former Marine. Now, the economy and jobs are atop voters' concerns. “Emotions are very high. People are very concerned about the economy and tiring of just limping along,” said John Ellinwood, a spokesman for the Kelly campaign. Kelly, 30, has called for lower taxes and more energy production as a way to improve the economy. He would roll back federal regulations and environmental protections in an effort to boost oil and gas drilling. Barber also is trying to convince voters that he understands their concerns. He frequently talks about building up the solar industry and the need to cut taxes, but only for the middle class. While Kelly has made it clear he would not support any income tax increases, Barber has said the wealthy need to “pay their fair share.” Immigration is still an important issue. Kelly wants a double-layer fence built along the district's border with Mexico. Barber is skeptical the fence would work on the district's rugged terrain. He has called for more manpower, horse patrols and the use of drones. The Tucson region is home to a growing population of retirees who rely on Medicare and Social Security. Kelly said in 2010 that privatizing the programs was a “must.” He said he would protect Social Security for current seniors but the program needed to be “phased out.” Giffords assailed his comments with great effect. Democratic groups have employed a similar game plan this election. “They're taking snippets from video out of context,” said Ellinwood. But Kelly has dialed back his rhetoric on Medicare and Social Security while still saying future Social Security participants should get a chance to opt out of the program. Both parties' national organizations have invested in the race. A win will give the victor a chance to claim momentum five months before November. A loss for Democrats would add to the difficulty of gaining the 25 seats they need to take control of the House. Republicans now hold a 240-192 advantage with three vacancies, including Giffords' seat. Tuesday's victory will be only a temporary one. Both candidates are promising to run for a full term in the fall, setting up a possible November rematch in a newly redrawn district that is friendlier to Democrats. Senate primaries in four states are expected to set up competitive races in November, which will help determine which party controls the Senate next year: -Former Sen. George Allen is the heavy favorite to win Virginia's Republican Senate primary, which includes three other candidates. The winner takes on Democrat Tim Kaine. -Six Republicans and four Democrats are running to fill the Senate seat being vacated by Republican Sen. Olympia Snowe in Maine. The front-runner, former Gov. Angus King, isn't on the ballot because he's running as an independent. -Republican Sen. Dean Heller and Democratic Rep. Shelley Berkley each are expected to prevail with ease against a slate of unknowns in Nevada. Their fall race would be one of the most competitive in the country. -In North Dakota, Rep. Rick Berg and businessman Duane Sand are vying for the Republican nomination in the race to replace retiring Sen. Kent Conrad. Democrat Heidi Heitkamp is running unopposed. In South Carolina, most of the interest is in the new 7th District in the northeastern corner of the state. Nine Republicans and four Democrats are running for the new House seat. Copyright 2012 The Associated Press. All rights reserved. This material may not be published, broadcast, rewritten or redistributed.