The investigation into the murder of Ashling Murphy remained focused on a man being treated for wounds in a Dublin hospital on Sunday night.
As reported in The Irish Times, the man's DNA was being cross-checked against a DNA profile taken from the crime scene.
It is understood his DNA is also being cross-checked against forensic evidence which was taken from the mountain bike believed to have been used by the killer.
Gardaí were alerted about the man's presence at a Dublin hospital after he arrived and could not satisfactorily explain the serious injuries he was being treated for.
After initial checks were made, links to Co Offaly were found.
On Sunday, Gardaí were searching the Grand Canal at Cappincur, Tullamore, Co Offaly, looking for a weapon which may have been used during Ms Murphy's murder on Wednesday.
A public facility in Co Offaly was also searched for a weapon at the weekend.
The suspect in the investigation has been treated in hospital for injuries since last Thursday.
The man, who has a partner and children, sustained a number of wounds, some of which are believed to be self-inflicted and serious.
On Sunday night, it was still unclear as to when the suspect would be mentally assessed before being interviewed by Gardaí. He has spoken briefly to Gardaí but has not been arrested.
Meanwhile, a substantial amount of evidence was gathered from two properties in south Dublin and in a village in Co Offaly over the weekend.
A vehicle which was searched and forensically examined is understood to be owned by a man close to the suspect.
Investigators are trying to determine if the suspect was in that vehicle, and the two houses, at any point after the murder.
Gardaí are also trying to determine if the suspect was in the Tullamore area last Wednesday and, if so, how he got to Dublin.
On Sunday, Ms Murphy's family visited the Grand Canal and walked the towpath towards the crime scene with gardaí.
The family described themselves as “heartbroken” in the notice setting out her funeral details.
Ms Murphy's remains are reposing at her family home just outside Tullamore and are due to be removed on Tuesday to arrive at St Brigid’s Church, Mountbolus, Co Offaly, for requiem Mass at 11am, with burial afterwards in Lowertown Cemetery.
Minister for Justice Helen McEntee pledged the Government’s new strategy on domestic and gender-based violence would have a clear goal of “zero tolerance” for any violence against women.
“What is being set out is four key pillars: protection, prevention, prosecution and policy co-ordination,” she said.
The strategy is expected to be published in March with an emphasis on education about violence against women in Irish society.