There are many results on graphs with the smallest eigenvalue at least -2. As a next step, A. J. Hoffman proposed to study graphs with the smallest eigenvalue at least -1 - √2. In order to deal with such graphs, R. Woo and A. Neumaier defined a new generalization of line graphs which depends on a family of isomorphism classes of graphs with a distinguished coclique. They proved a theorem analogous to Hoffman's, using a particular family consisting of four isomorphism classes. In this paper, we deal with a generalization based on a family H smaller than the one which they dealt with, yet including generalized line graphs in the sense of Hoffman. The main result is that the cover of an H-line graph with at least 8 vertices is unique.
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