The paper is about the edge covers of the graph. The change in the edge covering number of the graph is observed under the operations vertex removal, edge removal and edge addition. The necessary and sufficient condition is proved under which the edge covering number decreases when a vertex is removed from the graph also when an edge is added to the graph. The necessary and sufficient condition is proved under which the edge covering number increases when an edge is removed from the graph. Some results about the edge covering number have been obtained by considering the dual hypergraph of the graph under the operation vertex removal from the hypergraph