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Democratic state lawmakers have been mentioned as possible candidates, as has the name of Mrs. Giffords‘ husband, Mark Kelly, although he has publically quashed such speculation.

“That’s the great ‘mentioner’ out there, and there are going to be a lot of people mentioned,” said Arizona Democratic Party Chairman Andrei Cherny. “I think the best rule in situations like this is, ‘The folks who are talking don’t know, and the folks who know aren’t talking.’”

Those who decide to throw their hat into the ring will face yet another quirk in the race: The deadline to turn in nominating signatures for the general election comes before the special general election.

“I’m sure both parties and candidates of all stripes will in the days to come be thinking wide and hard about this district, and I’m sure there’s going to be a very vigorous contest,” Mr. Cherny said Sunday. “But today’s about thinking about a member of Congress who’s going to be irreplaceable no matter who wins that seat.”

Late Sunday night, Mrs. Giffords‘ office said that on Monday she will complete the meet-and-greet political event in Tucson that erupted in the shooting last year. Among those attending will be some of the wounded, those who helped them and those who subdued the gunman. She also will visit a family assistance center set up after she was shot, an event billed as her final act as a congresswoman in her district.

Mrs. Giffords also planned to attend President Obama’s State of the Union address on Tuesday in Washington.

But her political career may not be over, said a state Democratic party official who was among a group that met with her Sunday.

Jim Woodbrey, a senior vice chairman of the state party, said that at the meeting Mrs. Giffords strongly implied she would run again for office someday. He said the decision to resign came after much thought.

“It was Gabby’s individual decision, and she was not in any condition to make that decision five months ago,” he said. “So I think waiting so that she could make an informed decision on her own was the right thing to do.”

AP special correspondent David Espo contributed to this story from Washington.