Two unions representing 45,000 workers called a strike on Sunday when a labor contract expired and talks for a new contract failed after several weeks of negotiations.
While the two sides say they are still holding talks, the injunctions are just one sign of increasingly hostile relations outside the negotiating room. The Communications Workers of America, representing 35,000 workers, also has accused Verizon managers of injuring picketers.
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Verizon has sought injunctions to prevent "illegal" and "reprehensible" strike activities such as keeping managers covering for the striking workers out of buildings, according to company spokesman Rich Young.
Verizon said it received an injunction in Pennsylvania, filed for one in Delaware and vowed to seek injunctions in any state where strikers block buildings.
We did not condone violence of any kind: Union
The unions said in a joint statement on Tuesday that they did not condone violence of any kind and that they expected their members to follow the law.
Meanwhile the CWA said on Tuesday evening that it had received 23 reports of incidents where striking workers were hit or were narrowly missed by vehicles driven by Verizon managers or non-union contractors.
Young denied the accusations and said that "In some cases, union picketers are standing or throwing themselves in front of our vehicles."
Young referred to an Internet video where a person in a red shirt had a young lady, presumed to be his daughter, stand in front of a truck to stop it. A CWA representative declined comment on the video and Reuters could not verify its origin.
On a picket line in Brooklyn, New York, Hector Soto, 47, said union members picketing there were not blocking managers from entering the building to work.
"We're not stopping them physically, but we're letting them know they're scabs," said Soto, a shop steward and customer care representative for Verizon business clients.
"We're doing what we legally can do, which is protest," Soto said. Minutes later picketers could be heard yelling "scab."
The strikers are technical workers and customer service representatives for Verizon's U.S. northeast and mid-Atlantic wireline unit, which provides telephone and high-speed Internet services for homes and businesses as well as Verizon's FiOS television services.
Verizon is looking to cut costs in its wireline business
Verizon is looking to cut costs in its wireline business, which has been declining for years as customers get rid of home phones in favor of cellphones and Internet services.
But the unions say the company is asking for too many concessions in areas such as healthcare contributions, pensions, sick days and other work rules.
Verizon said it sent an update on Tuesday to lawmakers in the nine states affected by the strike and to leadership in the House and Senate and to Labor committees.
The statement reiterates the company's stance on issues such the need to reduce benefits, which it says total about $50,000 on average per employee.
It also argued that the CWA had agreed to similar changes with other companies such as AT&T Inc. The company said it spends $4 billion a year on healthcare for its employees. It has 196,000 employees including its wireless venture.
The company argued that employees need to work with it to adjust rules and policies in order for it to remain competitive in its Internet business.
"This is hard, but it is doable," Verizon said.
While the strike does not involve employees in Verizon Wireless, Verizon's venture with Vodafone Group Plc (VOD.L), some workers were picketing outside Verizon Wireless stores in New York on Tuesday. The wireless business depends on wired connections between wireless cell towers.
The International Brotherhood of Electrical Workers represents 10,000 of the striking workers.