Abdullah has boycotted the vote count, pitching Afghanistan into a political crisis as NATO combat troops withdraw from a 13-year war against Taliban insurgents.
The former foreign minister accused his poll rival Ashraf Ghani, outgoing President Hamid Karzai and election authorities of all committing fraud to deny him victory.
Reports of the ongoing vote count suggest that Ghani is well ahead in a surprise comeback after finishing behind Abdullah in the first-round election on April 5.
An early morning suicide attack in the city targeted Masoom Stanekzai, a senior official at the government body responsible for exploring peace talks with the Taliban.
Police said Stanekzai escaped the blast uninjured, but that one civilian was killed and three wounded.
The UN mission this week warned that if candidates "abandon the legal process and framework and appeal directly to supporters (it) could incite violence".
In a move that could lessen tension, Karzai yesterday backed Abdullah's calls for the UN to mediate an end to the deadlock.
"The disputes and doubts arising during the election process are a natural thing," Karzai added. "It is fine and calm in the country."
Abdullah's support is based among the Tajik minority and other northern tribes, while Ghani is a Pashtun -- Afghanistan's largest ethnic group, which is strongest in the Taliban heartlands of the south and east.
Abdullah faced Ghani in the run-off vote after the two came first and second in an eight-man first-round election, when Abdullah was ahead with 45 per cent against Ghani's 31.6 per cent.
As well as backing UN mediation, Karzai yesterday said negotiations between the poll rivals could be overseen by the two vice presidents.
"Now it is up to the candidates to choose one of these two options and the government to respect their choice and encourage them," he said.
The preliminary election result is due on July 2 and the final result, after adjudication of complaints, is scheduled for July 22.
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