It was proved recently by one of the authors that, if H is a path Pt (t>2 with t≠7 or 8) or an odd cycle Ct (t>3), then there is a unique maximal graph having H as a star complement for −2. The methods employed were analytical in nature, making use of the Reconstruction Theorem for star complements. Here we offer an alternative approach, based on the forbidden subgraph technique. In addition, we resolve the exceptional situations arising when H=P7 or P8.