Expansions for Reductions

Hall's theorem tells you when one side of a bipartite graph can be matched into the other. Its q-fold relative, the expansion lemma, says that when one side is at least q times larger, some chunk of the smaller side can be matched to q private neighbors each, and those neighbors go nowhere else. I show how such expansions turn into reduction rules. With q = 2 they tame high-degree vertices in Feedback Vertex Set. With general q they handle hitting all θ_q minor models (where θ_q is two vertices joined by q parallel edges), which gives polynomial kernels for this whole family of deletion problems.