An anonymous tip two days before Christmas led police to Kimberly and Kelly Yates and their mother, Elaine Yates, State Police Lt. Col. Joseph Philbin said. Elaine Yates, who had been living in Houston under the name Liana Lynn Waldberg, was arrested and faces arraignment Jan. 18 in Rhode Island.

Kelly Yates was 10 months old and her sister was 3 years old when they disappeared. Kelly Yates, now 32, and Kimberly Yates, now 35, weren’t living with their mother, who’s 69, but are in the Houston area and are in good health, Philbin said.

The girls’ father, Russell Yates, was told of the arrest immediately after it happened and is relieved, Philbin said. His contact information was given to his daughters.

“I’ve always been trying to find my children,” the father said. “Now at least, it’s up to them to get in touch with me.”

Russell Yates told The Providence Journal in a 1988 magazine story his wife had discovered he had been unfaithful to her and had threatened to move out in the middle of the night with the children.

He acknowledged punching her while they argued. A few weeks later, on Aug. 27, 1985, he came home after work around 2 a.m. and found them gone.

The case was featured on America’s Most Wanted, and police received tips from all over the country.

Investigators reviewed the case and spoke with Russell Yates at least once a year, Philbin said. But it wasn’t until they received the anonymous tip that they cracked the case.

Authorities said they used court databases, Facebook and driver’s license photos to identify the mother and daughters. He said they compared their photos with baby photos to help identify them.

Court records show Elaine Yates legally changed her name in Texas seven years ago. The judge’s order includes her correct birthdate and maiden and married names.

It says the Texas Department of Public Safety had examined the petition and found she had no recorded criminal history. At the time, a felony warrant for child snatching issued in 1988 was outstanding in Rhode Island.

The name change order was filed Aug. 26, 2009, 24 years to the day after she disappeared with her children.

Neighbors Jesse and Fran Vaughan, who live in the same condominium development as Elaine Yates in affluent West Houston, said she seemed like a nice person.

The Vaughans, who have lived at the complex for 17 years, said Yates was in charge of putting together the newsletter for the homeowner’s association but they weren’t aware she even had daughters.

“Come to think of it, I don’t know if anyone knew her that well,” Fran Vaughan said.

No one answered knocks on the doors of three addresses listed in records for Yates in West Houston and her daughters in Deer Park and Baytown.

Philbin wouldn’t comment on where Yates and her daughters had been since 1985 and said he didn’t know when they got to Houston.

He said he couldn’t comment on whether the daughters knew they had been taken or whether they knew their father was alive.

As for the mother, Philbin said she had been successful in getting away with her crime for 31 years.

He described her as being very cooperative with investigators since police knocked on her door. She was arraigned in Houston, waived extradition and was being transported to Rhode Island.

Russell Yates, asked if Elaine Yates should be prosecuted, said: “That isn’t going to help me, her or anybody else at this point. I just want to see my kids.”

It’s the second time in less than a week police announced they had solved an old child disappearance.

On Jan. 13, authorities in Florida charged a woman with kidnapping a newborn from a hospital 18 years ago and raising the child as her own.

The 18-year-old, Kamiyah Mobley, met her biological parents Saturday in Walterboro, South Carolina, where she was raised under a false name.

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By MICHELLE R. SMITH. AP writer Juan Lozano in Houston contributed