Police have been granted more time to question James Allen who went on the run after two people were killed in separate attacks.
The 36-Year-Old was arrested in Leeds on Sunday after he was spotted by an off-duty police officer.
He is being questioned by Cleveland Police following the murder last week of 81-year-old Colin Dunford in Middlesbrough and 50-year-old Julie Davison in Whitby, North Yorkshire.
Allen is being held at a police station in Middlesbrough on suspicion of the two murders and a sexual assault.
Cleveland Police confirmed it has obtained a warrant of further detention for the 36-year-old, but has not revealed how much longer officers have been given to question him.
The force has made an appeal for members of the public to check their properties for signs of anyone living rough.
Detectives have been trying to piece together Allen's movements over the last week and asked for anyone with information, or who believes they saw Allen, to contact them.
North Yorkshire Police said searches in and around the Whitby area were expected to continue throughout the week and asked people to check for any discarded property.
Detective Chief Inspector Steve Smith, who is leading the North Yorkshire Police investigation, said: "As we continue the inquiry into the murder of Julie Davison in Whitby, we are appealing to members of the public in the area to check their bins, gardens and out-buildings for any discarded personal items including footwear and clothing, and for signs of anyone living rough.
"People out and about in the rural areas around Whitby and Scarborough are also asked to look out for any discarded clothing or footwear.
"Anyone who does find any items or believes someone has been living rough on or near their property, the advice is not to touch or disturb them. Just leave them in place and call Cleveland Police immediately on 101."
The appeal came as Ms Davison's family released a statement speaking of their sadness at her death.
The statement read: "We are devastated by the sad loss of our lovely Julie.
"She was one of the kindest people you could ever wish to know and went out of her way to help others. She did not deserve to die in such terrible circumstances.
"We are so very sad that her hopes of building a new life back in her home town of Whitby ended so tragically."
The family thanked everyone who had sent them messages of support and asked to be allowed to grieve in peace.
Cleveland Police launched a murder investigation after Mr Dunford was found dead at his terraced house in Leven Street, Middlesbrough, last Monday.
A post-mortem examination on Mr Dunford, who was last seen alive at around 5pm last Sunday, showed he had suffered serious head injuries as a result of a significant assault.
Detectives said there was no sign of forced entry at the house.
Ms Davison, 50, who lived alone, was found dead by her sister at her rented flat in Church Square, Whitby, on Wednesday. She had also suffered serious head injuries, police said.
Allen, who is originally from Blackpool, is believed to have known Mr Dunford when he lived next door to him in Leven Street about a year ago but it is not thought Allen knew Ms Davison.
Allen travelled from Middlesbrough, where he was on Monday, to Whitby and Scarborough, where he was captured on CCTV images on Tuesday and Wednesday.
He is believed to have been in Leeds from Thursday, the day police found his mountain bike in Scarborough.