Detective Senior Sergeant Kallum Croudis said at a press conference this afternoon police were taking steps "around ensuring people cannot leave New Zealand".
“We have identified some witnesses who have given us some really good time related information, which semi-inform our understanding of when things occurred,” Det Sen Sgt Croudis said.
Mr Singh, 27, was found lying amid shards of glass outside his Hillary St home on Monday morning.
Inquiries had suggested this was a localised incident, and police were following ‘‘strong lines of inquiry’’, he said.
Det Snr Sgt Kroudis said evidence had suggested a sharp weapon as opposed to a shard of glass had been used.
He said it was ‘‘highly probable’’ there was a fight of some kind at the address but police were keeping an open mind with regards to a motive.
The broken glass at the property was ‘‘deformed’’ and this evidence was being used to inform their decision about the time of death.
Inquiries with friends of Mr Singh had revealed he had been concerned about items stolen from his garage.
He did not know if any of Mr Singh's possessions had been stolen.
There was ‘‘absolutely nothing’’ to suggest the incident was a hate crime or racially motivated.
Det Snr Sgt Kroudis said based on his injuries police believed he was killed on Sunday.
It emerged this week that Mr Singh had been excitedly sharing plans to holiday with his new wife the night police believe he was murdered.
Friends of Mr Singh, who shared pizza with him that night, told the Otago Daily Times they were in a state of shock after learning of the homicide investigation.
The friends, who like Mr Singh were migrants from India, said they never thought it was possible for something like this to happen in Dunedin and they felt scared.
Otago Punjabi Association Trust member Narindervir Singh said Mr Singh’s father was expected to fly in from India in the coming days.
While people had been desperate to help, the father had asked that no formal fundraising efforts start until he arrived.
Narindervir Singh said the scale of the show of support had been staggering.
“You won’t believe how many people have come forward,” he said.
“Not just Punjabis but the whole New Zealand community, all nationalities.”
He said the clear message to Mr Singh’s family was: “The community in New Zealand is with them. They’re not alone.”
