SIOUX FALLS, S.D. — There is a game politicians like to play with voters in this sparsely populated state. It starts with a question — What town are you from? — and ends when they find a mutual friend.

Elections are intimate in South Dakota, a state of 77,000 square miles but a population of 840,000, about the same as San Francisco. Parades loop twice through some communities, letting politicians pump every hand along one curb before crossing the street for a second lap. Candidates have been known to try soliciting votes from a stranger, only to find themselves, somewhat awkwardly, speaking to a relative of their opponent.

“If you don’t know someone, you know their neighbor,” said Mike Rounds, a former two-term Republican governor and the front-runner in the U.S. Senate race this November.

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Like everywhere, elections in South Dakota can get nasty, with negative ads flooding the cheap commercial airtime. But in this close-knit state, where retail politics involves living-room conversations and one-stoplight towns, attacking an opponent can be a delicate matter.

By consensus, Rounds leads three other candidates in a contest this year to replace Sen. Tim Johnson, a Democrat who is retiring. The South Dakota race is a presumed pickup for Republicans as part of their plan to take control of the Senate this fall. The uphill battle has underdog campaigns confronting a tough political calculus: In a state where the degrees of separation may number only two, how forcefully can a candidate criticize a front-runner before it backfires?

“There’s a line you can’t cross,” said Tom Daschle, the former Senate majority leader from South Dakota, who added that the final weeks of a race often turn ugly. “That line constantly is changing, but people view it as a pretty low bar.”

Few people know this better than Rounds. In 2002, he was a former majority leader of the state Senate and polling third in the Republican primary race for governor behind Mark Barnett, the attorney general at the time, and Steve Kirby, a former lieutenant governor. Both were better known and better funded than Rounds, who was written off by pundits.

Then the mudslinging started between the better-known candidates. Barnett criticized Kirby for investing in a company that used cadaver tissue for cosmetic surgery, saying it contributed to a shortage of skin for burn victims. Each campaign pummeled the other in television commercials and at news media events.

“As soon as we went negative, our volunteers evacuated,” said Jason Glodt, who worked for the Barnett campaign and is now with Rounds. A month before the election, he said, only one volunteer showed up at a campaign event to go door-to-door with Barnett in Rapid City.

Rounds was rewarded for avoiding the fracas, winning the three-way primary by double digits and coasting to a November victory.

And yet the lesson was more complicated than it appeared.

Two years later, Democrats say Daschle failed to hit back hard enough to fend off attacks by John Thune, a Republican, and outside groups. The most expensive race in South Dakota history, the heated battle became a referendum on the three-term senator, whom Thune criticized as having lost touch with South Dakotans and, among other things, for the lobbying activities of Daschle’s wife, Linda. The most damning 30-second spot, titled “In his Own Words,” showed Daschle calling himself a Washington resident.

Thune’s upset victory was proof that personal attacks work in South Dakota, under the right circumstances, strategists say.

But in 2008, when Johnson was recovering from a brain hemorrhage and declined to appear in debates, voters rejected a Republican challenger who questioned “whether he can hold his own.” Despite Johnson’s slow speech and use of a wheelchair, he won with ease.

“If they can’t win it without knocking their opponent down, they shouldn’t be running,” said Albert Albers, 66, of Lennox, echoing a sentiment heard from voters around the state.

In the Senate race this year, opponents contend that Rounds, who won a five-way Republican primary with 55 percent last month, may not be as formidable as political experts predict. Their success in the coming months may depend on finding ways to criticize him without going too far.

For the past year, Rick Weiland, the Democratic nominee and a former aide to Daschle who ran unsuccessfully for Congress in 1996, has barnstormed the state, introducing himself to voters in every town and trying to make up enough ground to persuade his party in Washington not to abandon South Dakota for more competitive races.

Weiland has repeatedly accused Rounds, who said last year that he planned to raise $9 million for his Senate bid, of “shaking down big money” from outside donors and of trying to buy the election. He rewrote the lyrics to the classic country tune “I’ve Been Everywhere” to throw another jab on the airwaves: “I said you can raise all your millions by the sack,” he sings, clutching an acoustic guitar in his latest television ad. “The time has come for us to take our country back.”

Voters seem to enjoy the musical approach; Republicans stopped Weiland on a recent afternoon of campaigning to say his cover of the song, most famously sung by Johnny Cash, was pleasantly stuck in their heads.

“I think you can deliver medicine if you serve it with a little sugar,” Weiland said in an interview, calling his criticism of Rounds fair game. “I don’t have an ad to waste.”

He plans to perform at the state fair next month and may release more songs, including another Cash revamp — “I Draw the Line” — that again hits Rounds on contributions.

Rounds’ campaign finance reports show a mix of donors from South Dakota and supporters outside the state.

Other candidates, too, have the front-runner in their sights. Larry Pressler, a Republican former senator who changed the dynamics of the race by running as an independent, recently criticized Rounds for not objecting more to a resolution by the state Republican Party last month that called for the impeachment of President Barack Obama.

Another independent candidate, Gordon Howie, a former state legislator with ties to the Tea Party, has called Rounds a career politician and not a true conservative.

Now, after a hostile primary, Rounds is trying again to avoid the fray and says he has fought with party operatives who think he should more aggressively define his rivals. He will face other candidates for the first debate of the general election at Dakotafest, an agricultural trade show, in August.

“He hasn’t been punched in the face yet,” said Nick Miner, a communications aide for Weiland. “Tone is important, but at some point you have to engage.”