The Gorkha Janmukti Morcha (GJM), spearheading the agitation, said a separate state was the "aspiration" of the people in the hills and it should be respected.
"The Centre should open a dialogue and take concrete steps," GJM general secretary Roshan Giri told reporters here.
He said the Mamata Banerjee government was dealing with the situation as a law and order problem.
The GJM leader said, "We believe in democratic movement. We want a dialogue. Why a dialogue cannot be opened?"
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The police said a vehicle was set ablaze in Darjeeling's Six Mile village and two cars were damaged in Kurseong by the protesters this morning.
The GJM activists held demonstration here in Chowkbazar in which children in traditional dress were also seen carrying 'We Want Gorkhaland' placards.
While Internet services remained suspended, the police and the security forces patrolled the streets and kept a tight vigil on all entry and exit routes.
Meanwhile, the boarding schools in Darjeeling, which were scheduled to reopen tomorrow, have extended their vacation due to the indefinite shutdown.
The vacation had begun on June 23 when the GJM had given a 12-hour window to the schools to evacuate the boarders.
"We have decided to extend the vacation as the shutdown is going on. We have not given any time frame as to when the schools will reopen. As soon as normalcy returns in the hills, we will inform parents and reopen the schools," Principal of St. Joseph's North Point, Father Shanjumon told PTI.
Normal life was affected due to the shutdown which entered its 19th day.
Except pharmacies, shops, schools, colleges, restaurants, private offices and hotels remained closed in the hill resort which has been on the boil since June eight.
While the West Bengal police has said that two persons have died in the violence, the GJM has claimed two more of its supporters were killed on June 17.
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