Angela Humphrey's body was found at her home by neighbours in Barmulloch, Glasgow, on Wednesday, almost a week after she was last seen.
She is closely related to Sean Humphrey, 26, who was found dead at his flat in Shettleston last October. Two men have been arrested in connection with the death.
Police said the 48-year-old grandmother, known to friends as Angie, was the victim of a violent attack.
Detective Inspector Allan Burton stressed that Ms Humphrey's death was an isolated incident and not related to any other investigation.
Relatives raised the alarm after they spotted her body in the living room of her ground floor flat.
Asked if he had a message for the killer, Mr Burton said he was confident his officers would soon be knocking on the person's door.
Ms Humphrey had moved into the property after separating from her second husband last year. She had been living there for just two weeks when she was killed.
She had two daughters who are believed to be in their 20s.
Mr Burton said: "Angela's family are devastated. One of her daughters is pregnant and the other has a young child. She had a future with her family and it's been taken away."
Her daughter's distraught partner and a family friend smashed the bedroom window to get in to the property, but it was too late.
Detectives are examining every aspect of Ms Humphrey's life as part of the wide-ranging inquiry.
Closed-circuit camera footage from the area is being examined in an attempt to piece together the victim's last movements.
Ms Humphrey was last seen in the Gallowgate area of the city by relatives on February 21.
Mr Burton believes local revellers and taxi drivers may hold the key to where she went next.
He added: "Last Friday she was in Gallowgate and Parkhead, socialising in bars in the afternoon with family members.
"We believe she returned home in the evening, but it is not known how she made her way home. We need to confirm exactly how she got home. If you saw Angie, please get in touch.
"I would appeal to any taxi drivers who picked up a passenger that matches Angie's description to get in touch. We need to build up a picture of her movements."
Police Scotland's Major Investigations Team is attempting to establish the motive for the killing. Detective Inspector Burton declined to reveal if a weapon was used, but said he did not believe robbery was a motive.
He said: "It is possible the person responsible may have been injured, they may have scratches to their face.
"Or they might be acting in a strange manner. Please contact police if you have any concerns about family members or anyone you know."
Anyone with any information is asked to contact the police incident room on 0141 532 3856.