The CPI(M) complained to the state Election Commission about Howrah Mayor, Mamata Jaiswal, suffering an injury in a bomb explosion for which she was hospitalised.
The Howrah Municipal Corporation is a CPI(M)-controlled body.
A senior officer of the state Election Commission said, "We have received a complaint from the CPI(M) ... We are verifying it."
The police said a bomb exploded in front of the car of the Mayor in ward number 13 near St Agnes school.
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Trinamool Congress leader and Irrigation minister Rajib Banerjee said, "If that is so, why is the CPI(M) not registering an FIR?"
He said that the allegation was to tarnish TMC's image and get sympathy, since the CPI(M) was losing political ground after being rejected by voters in every election.
In Salt Lake Municipal Corporation, there was report of a clash between the TMC and CPI(M) supporters at FD block concerning a by-election at ward no 13. The police brought the situation under control.
The election is being viewed as another test of the popularity of the Trinamool Congress-led government after the TMC received an overwhelming mandate in the panchayat elections in July this year.
The main contestants in the civic polls are TMC, Congress, CPI(M)-led Left Front and the BJP.
Polling is being held amid tight security in Howrah Municipal Corporation and in Jhargram, Midnapore, Krishnagar and Behrampore municipalities.
While the CPI(M) controls the Jhargram municipality and the Howrah Municipal Corporation, Trinamool runs the Midnapore and Krishnagar civic bodies. The Congress controls the Berhampore municipality.