Let $X$ be a weak del pezzo surface, which means that the anticanonical bundle $-K_X$ is nef. There is a classification of such surfaces by the configurations of its $(-2)$-exceptional components. I wonder whether there exists some classification of rational surface which is non weak del pezzo,i.e: the surface might admit $(-n)$-curves such that $n\geq 3$. For example, consider a surface $Y$ such that $(-K_Y).C\geq -1$, did anybody do some work to classify them? I expect some work like classifying them by looking at different configurations of $(-2),(-3)$-curves just like the case in weak del pezzo surfaces.

Thanks