For the first several months of President Obama's second term, the newly reelected chief executive got no gentle treatment from Lindsey Graham, the senior Republican senator from South Carolina.

Senator Graham savaged UN Ambassador Susan Rice, a close friend of Mr. Obama's, for her public handling of the terror attack in Ben­ghazi, Libya. He excoriated a onetime colleague, former Sen. Chuck Hagel (R) of Nebraska, when the president put up the taciturn Midwesterner as his choice for secretary of Defense. And he balked vociferously at the White House's push for more stringent gun laws.

So whom did Obama call to organize a dinner for himself and often-recalcitrant Senate Republicans as the first serious bipartisan outreach of his second term?

Graham, of course.

Over nearly two decades in Congress, Graham has earned almost singular renown for his political and policy entrepreneurship, a consistent willingness to find conservative ways to solve problems rife with partisan divisions. The bigger the problem, the uglier the politics, the more likely the slight man with the classic Southern-lawyer demeanor will be at the center of the fray.

Sometimes, as with Ms. Rice and Mr. Hagel, it's as a partisan slasher. But in other cases, from offering fixes to Social Security under President George W. Bush to pushing for comprehensive immigration reform to supporting a "grand bargain" on fiscal issues, Graham has shown an equal fidelity to finding solutions in politically uncomfortable places.

Those who have followed Graham see his tenacity in trying to work things out rooted in his personal history. While en route to becoming the first in his family to graduate from college, Graham lost both of his parents in a 15-month span. He became the legal guardian of his 15-year-old sister when just a junior at the University of South Carolina.

"A couple of times we tried, with others, to develop Social Security reform," says former Sen. Joe Lieberman (Ind.) of Connecticut, a longtime Graham ally. "He said to me, privately, 'I'm never going to be part of hurting Social Security because if it wasn't for Social Security, we wouldn't have made it....' As he looks back at his life, I think there's a voice inside his head that says, 'I can't believe it. Now that I'm here, I've got to make it count.' "

Graham kept moving forward – into the Air Force as a lawyer and then, more than a decade later, into Congress with the backing of South Carolina's political godfather, the late Sen. Strom Thurmond (R). But it was that experience of vulnerability and survival that helped shape how he operates in the Senate today.

"I think he's realized he needs a lot of people to help him, and people of all different stripes have helped him as he's gone through life," says David Woodard, a political science professor at Clemson University in South Carolina, who ran two of Graham's House campaigns in the 1990s. "I believe he thinks he can talk and work with a lot of people that most people don't think they can work with."

Graham, who is fiercely loyal to his friends, is able to sit down with people ranging from liberal Sen. Chuck Schumer (D) of New York to members of the three local South Carolina Republican committees who expressed no confidence in him in 2009 for not being conservative enough.

Graham remains under fire from some activists on the right, who see him as too willing to compromise on core principles. Even before the 2012 election, groups like the Club for Growth and the Senate Conservatives Fund were open about their interest in finding a candidate to challenge him in 2014.

In that sense, Graham is a test for whether skillful politicians can reach bipartisan compromise and survive. "The rest of the country is looking to see what Lindsey Graham is going to do [on immigration]," says Jeremy Robbins, director of the Partnership for a New American Economy, a group partially funded by New York City Mayor Michael Bloomberg.

Mr. Robbins's group and Republicans for Immigration Reform, a new "super political-action committee" dedicated to supporting conservatives who take tough votes on immigration, are running advertisements to counter negative spots aired by critics of Graham in South Carolina.

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If he fails? The man whom conservative radio talk-show host Rush Limbaugh once derided as Lindsey "Grahamnesty" – a derisive term for his willingness to support some sort of legal status for illegal immigrants – could lose to a more ideologically pure lawmaker in a primary next year.

And if he succeeds? "If he can become part of [a big compromise] that passes and survive," says Julian Zelizer, a congressional historian at Princeton University, "then I think a lot of Republicans would think twice about their fears about entering this kind of negotiation."