AbstractThe aim of this paper is to study the Lovász‐Schrijver PSD operator applied to the edge relaxation of the stable set polytope of a graph. We are particularly interested in the problem of characterizing graphs for which generates the stable set polytope in one step, called ‐perfect graphs. It is conjectured that the only ‐perfect graphs are those whose stable set polytope is described by inequalities with near‐bipartite support. So far, this conjecture has been proved for near‐perfect graphs, fs‐perfect graphs, and webs. Here, we verify it for line graphs, by proving that in an ‐perfect line graph the only facet‐defining subgraphs are cliques and odd holes.