Induced Erdős--Pósa property for long holes, long thetas, and beyond

What the math gives to ML

The paper gives a constructive packing-versus-hitting principle for induced long cycles and theta graphs: either many mutually anti-adjacent forbidden patterns exist, or a small closed neighborhood N[X] intersects every such pattern, with |X| = O(tk log k). Its transferable asset is not the graph-specific object itself, but the conversion of complicated global interaction graphs into small vertex-neighborhood separators and recursively anti-adjacent components. This suggests a separator-aware sparse attention architecture in which a learned token-interaction graph is periodically decomposed by removing neighborhoods of a small set of separator tokens, yielding independent attention blocks and reduced KV memory. The theorem can serve as a structural target and benchmark for a practical greedy separator heuristic, even though exact induced-minor detection should be replaced by bounded-length pattern search.