Conjecture on minimally _g-imperfect graphs

Let $G$ be a graph. It is minimally $\gamma_g$-imperfect if $G$ is not $\gamma_g$-perfect, while every proper induced subgraph of $G$ is $\gamma_g$-perfect. The anti-holes $\overline{C_n}$ for $n\geq 5$, together with the eight additional graphs listed in Proposition~, are the graphs under consideration. Classification conjecture. There are no other minimally $\gamma_g$-imperfect graphs but those listed in Proposition~. This conjecture is based on computer verification for graphs with at most nine vertices and on the paper's structural results for connected and triangle-free imperfect graphs; its status beyond those verified cases remains open.