Motivated by the Green-Griffiths and Lang-Vojta conjectures, it is expected that the algebraic exceptional set of a log-surface $(X,B)$ of log-general type - which parametrizes rational curves on $X$ meeting $B$ in at most two points - is finite. In this talk, I will discuss recent results on this set for the cases where $X$ is the projective plane or a Hirzebruch surface, and $B$ is a curve with three irreducible components. This talk is based on [arXiv.2507.13280] and work in progress.