On Collusion-Free Teaching in Planar Graphs

Non-clashing teaching minimizes teaching complexity among models satisfying the Goldman--Mathias collusion-avoidance criterion. While the Vapnik--Chervonenkis dimension (VCdim) measures the complexity of learning from samples, non-clashing teaching dimension (NCTD) measures the number of examples required when a teacher can select them. Whether NCTD is always bounded above by VCdim remains a fundamental open question in learning theory. In this paper, we investigate this question for the concept classes of closed neighborhoods in planar graphs. We prove a tight upper bound of two on their NCTD, thereby establishing the conjectured inequality for this family. For the positive variant NCTD+, in which teaching sets contain only positive examples, we obtain a tight upper bound of four. Finally, we show that deciding whether either NCTD+ or NCTD is at most one is NP-complete, even for concept classes of closed neighborhoods in planar graphs. In particular, despite the upper bound of two on NCTD for such classes, distinguishing between NCTD values of one and two remains NP-hard.