Planar-graph teaching needs just 2 examples: the old bound was 5, and telling 1 from 2 stays NP-hard
For concept classes of closed neighborhoods in planar graphs, the non-clashing teaching dimension is now bounded by 2, and by 4 in the positive-only variant, with both bounds tight; the previous bounds were 5 and 7. The proof confirms the conjectured inequality between teaching dimension and VC dimension for this family, yet deciding whether either dimension is at most 1 remains NP-complete, and distinguishing the values 1 and 2 remains NP-hard.