The Brexit vote in the EU referendum has triggered a renewed debate in Northern Ireland on its constitutional position within the UK.
With the region backing Remain by 56% to 44%, Sinn Féin has insisted the time has come for a border poll on Irish unity.
The call has been rejected by the Democratic Unionists who have claimed Northern Ireland's status as part of the UK was "safe and secure".
In recent years the thorny constitutional issue has dropped down the political agenda at Stormont, as more focus was placed on economic and social affairs. But the result of the referendum - with the UK overall voting leave, but Northern Ireland voting to stay - has propelled the question back to public prominence.
A border poll can only be called by the region's Secretary of State in circumstances where there is clear evidence of a public opinion swing towards Irish unity.
Northern Ireland Secretary Theresa Villiers said she did not believe the criteria for triggering a border poll had been met.
In Northern Ireland 440,707 (56%) people voted Remain and 349,442 (44%) Leave.
Of the North's 18 constituencies, 11 voted Remain and seven voted Leave.
Sinn Féin Deputy First Minister Martin McGuinness said: "For us to be dragged out of the EU against our will is absolutely unacceptable.
"We have been calling for a border poll prior to this referendum. The outcome absolutely strengthens our argument, given that the democratically expressed wishes of the people of the North are going to be effectively ignored by a negotiation which will take place in all probability with a new British Prime Minister."
Mr McGuinness, who was flanked by party president Gerry Adams as he reacted to the result at Stormont Castle, said those who believed lost EU monies could be replaced were living in "cloud cuckoo land".
However, Stormont's First Minister Arlene Foster rejected the Sinn Féin demand.
"The call for a border poll was as predictable as the flowers in May," she said. "We knew it would come but the test has not been met so therefore I don't believe it will happen."
The DUP leader added: "I think we are now entering a new era of an even stronger United Kingdom.
"I am very much looking forward to working with our colleagues in our national government to build a very strong, outward-looking UK moving forward and of course Northern Ireland as a very key constituent part of that."
The renewed focus on Northern Ireland's constitutional position came as Scotland's First Minister Nicola Sturgeon said a second referendum on Scottish independence was "highly likely".
'More serious issues'
Taoiseach Enda Kenny said there was no evidence of a shift in the Northern Ireland electorate for a border poll on a united Ireland.
"That [provision for a border poll] is contained in the Good Friday Agreement, provided that the Secretary of State of the day considers that there would be a serious movement of a majority of people to want to have a situation where they would join the Republic," he said.
"There is no such evidence."
Mr Kenny added: "There are much more serious issues to deal with in the immediate terms and that is where our focus is."
Foreign Affairs Minister Charlie Flanagan also dismissed the idea of a poll.
"I believe any further referendums, with particular reference to the status of Northern Ireland, or otherwise, would in the circumstances be most unhelpful," he said.
A UK Government spokesman said: "Although there were strong views on both sides of the debate, all the main NI parties are agreed on one thing. The people of NI want to see peace and political stability continue, they want their elected representatives to work together, and in co-operation with the UK and Irish Governments.
"The UK Government is strongly committed to the Belfast Agreement and the institutions it created. Nothing in this vote will undermine the workings of the devolved government, the North/South institutions or the British-Irish Council.
"The UK Government will continue to abide by its commitments in the Belfast Agreement. But there is no reason to change our view that a majority of the people of NI support the current political settlement and want to remain part of the UK. As the Government made clear in its NI manifesto last year, it continues to be the case that the requirements in the Belfast Agreement for a border poll are not met."