"There are no words that can possibly express how we feel," the family of Jared Loughner said in a statement handed to reporters outside their home. "We don't understand why this happened. It may not make any difference, but we wish that we could change the heinous events of Saturday. We care very deeply about the victims and their families."
The family's statement came as Tucson residents prepared for a community memorial service Tuesday night and a visit from the president a day later.
Earlier, a neighbor described the parents as devastated and guilt-ridden. Jared Loughner's mother has been in bed, crying nonstop since the shooting rampage on Saturday, neighbor Wayne Smith, 70, told KPHO-TV. Amy and Randy Loughner want to know where they went wrong with their son.
The younger Loughner is charged with trying to kill the Democratic lawmaker and killing a federal judge.
"I told them they didn't fail. They taught him everything about right and wrong," Smith said. "We all know you can teach someone everything and have no control how it works out."
At University Medical Center, Giffords remained in critical condition, but doctors said she is able to draw breathes on her own. Dr. Michael Lemole, Giffords' neurosurgeon, said, however, that doctors left the breathing tube in Giffords to protect her airway.
When asked about swelling in her brain on the third day, which is when it often reaches its peak after an injury, Lemole said a CT scan early Tuesday showed no increase in swelling. But he cautioned that it can sometimes take longer for brain swelling to reach its peak.
Meanwhile, the parents of one of Loughner's close friends, Zach Osler, described a troubled relationship between the suspect and his parents. He is their only child.
In an interview with The Associated Press, George and Roxanne Osler recalled the only time they met the rest of his family.
In 2008, the Loughner parents showed up at the Osler's doorstep looking for their son, who had left home about a week before and broken off all contact, George Osler IV said.
Jared often spent time at the Osler's place, sometimes watching conspiracy-theory movies with Zach and his younger brother, George V.
Mrs. Osler said she was struck by how unfailingly polite he was -- far more so than their son's other teenage friends.
With the Loughners at his house, Zach Osler told them the name of the local hotel where their son was staying. The Loughners were able to patch things up and Jared moved back in with his parents, Zach's father said.
After that, the elder Osler sometimes would see Mrs. Loughner at the local supermarket, though they didn't chat much. He recalled that every time he saw her she had at least one 30-pack of beer in her cart.
Jared Loughner grew up on a typical Tucson block of low-slung homes with palm trees and cactus gardens out front.
Loughner's father moved into the house as a bachelor, and eventually got married and the Loughners had Jared, longtime next-door neighbor George Gayan said. Property records show Randy Loughner has lived there since 1977.
Gayan said he had "differences of opinion but nothing where it was radical or violent." He declined to provide specifics.
"As time went on, they indicated they wanted privacy," Gayan said. Friends of Jared Loughner have described him as a loner.
Unlike other homes on the block, the Loughners' is obscured by plants.
Amy Loughner got a job with the county just before Jared was born, and since at least 2002 has been the supervisor for Roy P. Drachman Agua Caliente Park on the outskirts of the city. She earns $25.70 an hour, according to Gwyn Hatcher, Pima County's human resources director.
Randy Lougher apparently has not worked for years -- at least outside his home. One thing he did do was fix up cars. Gayan said he had three "show cars" and two of Jared Lougher's friends said he bought a junker '69 orange Chevrolet Nova and made it pristine.
On Monday, Jared Loughner appeared in court wearing a beige prison jumpsuit and handcuffs and sporting a pink gash on the hairline of his shaved head. A judge asked if he understood that he could get life in prison -- or the death penalty -- for killing federal Judge John Roll.
"Yes," Loughner replied.
Loughner was being held without bail.
Meanwhile, a Mass for all the victims was set for Tuesday at St. Odelia's Parish in Tucson, and President Barack Obama was scheduled to arrive in Arizona on Wednesday for a memorial service days after calling the attack a tragedy for the entire country.
Loughner's court appearance in Phoenix on Monday gave the nation a first look at the man authorities say is responsible for the shooting that also left 14 injured outside a Tucson supermarket where Giffords had set up a booth to hear the concerns of constituents.
Eric Fuller, one of the survivors, said Tuesday on the CBS "The Early Show" that he felt the bullet that hit his knee but didn't know he had also been hit in the back.
"Not wanting to leave the world very soon, and not thinking that I could do very much except maybe get killed trying to stop him from the vantage point that I was at, I fell to the ground, as other people were doing -- and expected the worst to occur after that," he said.
Loughner is charged with one count of attempted assassination of a member of Congress, two counts of killing an employee of the federal government and two counts of attempting to kill a federal employee. Those are federal charges.
His newly appointed lawyer is Judy Clarke, who defended "Unabomber" Ted Kaczynski.