The mother of Adrian Dunne, who died along with his wife and two young daughters at their home in Monagear, Co Wexford, has been explaining why an undertaker became concerned for their safety.
It has emerged Adrian and his wife, Ciara, went to the undertaker with their children on Friday and specified what coffins they wanted to be buried in.
Speaking to RTÉ News, Mary Dunne said the undertaker became concerned when the girls' mother asked for two white coffins.
The results of post mortem examinations on the four bodies are expected to be known later today.
The Government has said it will set up an independent investigation into the circumstances leading up to the deaths of the Dunne family.
Speaking on RTÉ Radio's News at One, the Minister for Children, Brian Lenihan, said the matter had been discussed at length at Cabinet today.
He said the Minister for Justice, Michael McDowell, had asked for a report from An Garda Síochána, and he himself had asked for a report from the Health Service Executive.
Mr McDowell said that a number of issues have to be looked at in relation to the deaths.
He cautioned against jumping to conclusions in what he described as a very complex case.
Mr Lenihan said the terms of reference of an independent investigation could only be decided when the facts had been established.
Mr Lenihan said the circumstances of this case differed to a similar case in Co Wexford two years ago, because in this case no member of the family had sought assistance from the authorities.
The bodies of 29-year-old Adrian Dunne, his wife Ciara, aged 24, their daughters, Leanne and Shania, aged five and three, were found in their home yesterday afternoon.
The garda investigation into the deaths is continuing with a number of people being interviewed about the sequence of events over the weekend.
It is believed that no social worker from the HSE visited the Dunne family over the weekend and while a garda car drove past the Dunne family home a number of times, gardaí never called in.
The Garda Commissioner has appointed Superintendent Pat Mangan of Kilkenny Garda Station to inquire into the handling of the matter by the gardaí.
Healthcare officials unaware of risks
Healthcare officials have said they were unaware of any risks to the family of four.
Speaking on RTÉ Radio's Morning Ireland, the HSE's local health manager for Wexford, Pauline Bryan, said there were no known risks.
Mrs Bryan confirmed that the HSE's childcare manager was contacted on Saturday afternoon by gardaí.
She said the advice of the childcare manager to gardaí was for them to go to the emergency family doctors out of hours service, Caredoc, if they had concerns.
Gardaí were advised to take the children to a place of safety, such as Wexford General Hospital, if they were concerned about their welfare.
Mrs Bryan also said that the family were not on the 'at risk' register.
This morning the HSE in the southeast confirmed it was in contact with the Dunne family as recently as midday last Friday.
Last night, the Irish Society for the Prevention of Cruelty to Children called for an independent child protection investigation into the circumstances surrounding the tragedy.
The funeral arrangements for the young family are now being put in place.
Increase in calls to Samaritans after tragedy
Samaritans Ireland Director, Susanne Costello, has said that there has been a large increase in calls to the Samaritans over the past 24 hours, since news of the tragedy broke.
The Samaritans called on other service providers helping those at risk of self-harm to recognise that their services are needed around the clock and not just during normal working hours.
Ms Costello said the majority of calls to the Samaritans relating to emotional distress come at weekends and in the early hours of the morning.
She said that it was time other agencies recognised this.
She said the Samaritans would encourage anyone feeling effected by the issues to contact them on 1850-609090.
