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Irish police are continuing to hunt for the killer of Ms Murphy, who was found dead after going for a run on the banks of the Grand Canal in Tullamore, Co Offaly.
The Garda said it had made “significant progress” in its investigation but were not releasing details for operational reasons.
It is understood that gardai have identified a new person of interest, who is currently in hospital, and are waiting to speak to him.
Park Run runners in the Republic of Ireland, Northern Ireland and beyond held moments of silence on Saturday morning for Ms Murphy.
Hundreds of people also gathered in Cork on Saturday morning for a vigil, with more planned in towns and villages across the weekend.
A vigil will also be held in London on Saturday afternoon.
Thousands of people gathered in the late afternoon in Tullamore, Dublin, and Belfast on Friday, as Ireland continues to reel from the murder of Ms Murphy.
— Garvagh Forest parkrun (@GarvaghFparkrun) January 14, 2022
Ms Murphy’s family attended a candlelit vigil near the murder scene on Friday evening.
At the event, her father Ray Murphy paid a poignant tribute to the talented young musician by performing her favourite song on the banjo.
He broke down in tears while playing the final chords of When You Were Sweet Sixteen.
— Cabinteely parkrun (@cabinteelypkrun) January 14, 2022
Mr Murphy, along with his wife Kathleen and daughter Amy walked on the opposite side of the canal to where his daughter was assaulted and died on Wednesday.
Beautiful but heartbreaking singing at Ashling's vigil in Cork this morning.
I welled up when I saw the crowds walk from town as I cycled. Every one of them has a story, every one of them has a friend. This tragedy has touched us all pic.twitter.com/3tJD5KciGE
— Eoghan Ua Laoghaire Mac Giolla Phádraig 🇵🇸 (@eoghanolf) January 15, 2022
Taoiseach Micheal Martin has said that the murder has “united the nation in solidarity and revulsion”.
“No stone will be left unturned in terms of bringing this investigation to a completion and to bring the person responsible for this to justice,” he said on Friday.
Politicians have promised that all resources necessary will be provided to the gardai to find the killer.
The death of Ms Murphy has sparked fresh debate about the safety of women in Ireland, with many asking how such an attack could happen in broad daylight.
“We, as a society, need to face up to this. There is an epidemic of violence against women. It’s been going on for millennia, quite frankly,” Tanaiste Leo Varadkar said on Friday.