Hypertree-depth and minors in hypergraphs

We introduce two new notions for hypergraphs, hypertree-depth and minors in hypergraphs. We characterise hypergraphs of bounded hypertree-depth by the ultramonotone robber and marshals game, by vertex-hyperrankings and by centred hypercolourings. Furthermore, we show that minors in hypergraphs are ‘well-behaved’ with respect to hypertree-depth and other hypergraph invariants, such as generalised hypertree-depth and generalised hyperpath-width. We work in the framework of hypergraph pairs (G,H), consisting of a graph G and a hypergraph H that share the same vertex set. This general framework allows us to obtain hypergraph minors, graph minors and induced graph minors as special cases.