The edge clique cover number $ecc(G)$ of a graph $G$ is the size of the smallest set of complete subgraphs whose union covers all edges of $G$. It has been conjectured that all the simple graphs with independence number two satisfy $ecc(G)\leq n$. First, we present a class of graphs containing edges difficult to cover but that satisfy the conjecture. Second, we describe a large class of graphs $G$ such that $ecc(G)\leq \frac{3}{2}n$. This class is easy to characterize.