Iraqi Kurdish leader Massud Barzani hailed victories over the Islamic State jihadist group during a visit on Sunday to Mount Sinjar, which had been besieged by the militants for months.
Thousands of the autonomous Kurdish region's peshmerga fighters launched a major operation backed by US-led air strikes on Wednesday which broke the second IS siege of Mount Sinjar this year.
The Kurdish offensive threatens the links between the city of Mosul, the main IS stronghold in Iraq, and territory the militant group controls in neighbouring Syria.
"During the past 48 hours, the peshmerga opened two main routes to Mount Sinjar," Barzani said, adding: "We did not expect to achieve all these victories."
In addition to breaking through to the mountain, "a large part of the centre of the town of Sinjar was also liberated," he said, referring to the district's main settlement to the south of the mountain.
The Kurdish regional president said the peshmerga might participate in an operation to retake Mosul itself.
"We will take part if the Iraqi government asks us, and of course we will have our conditions," he said, without specifying what they would be.
IS spearheaded a sweeping offensive that has overrun much of Iraq's Sunni Arab heartland since June, presenting both an opportunity for territorial expansion and an existential threat to the country's Kurdish region.
Multiple Iraqi divisions collapsed in the early days of the advance, clearing the way for the Kurds to take control of a swathe of disputed northern territory that they have long wanted to incorporate into their autonomous region over Baghdad's objections.
But after driving south towards Baghdad, IS then turned its attention to the Kurds, pushing them back towards their regional capital Arbil in a move that helped spark US air strikes against the jihadists.
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Iraq Kurd chief visits Mt Sinjar after siege broken