We prove that for every planar graph X of treedepth h, there exists a positive integer c such that for every X-minor-free graph G, there exists a graph H of treewidth at most f (h) such that G is isomorphic to a subgraph of H ⊠ Kc. This is a qualitative strengthening of the Grid-Minor Theorem of Robertson and Seymour (JCTB, 1986), and treedepth is the optimal parameter in such a result. As an example application, we use this result to improve the upper bound for weak coloring numbers of graphs excluding a given graph as a minor.

Abstract. Alon, Seymour, and Thomas [ J. Amer. Math. Soc., 3 (1990), pp. 801–808] proved that every [Formula: see text]-vertex graph excluding [Formula: see text] as a minor has treewidth less than [Formula: see text]. Illingworth, Scott, and Wood [ Product Structure of Graphs with an Excluded Minor, preprint, arXiv:2104.06627 , 2022] recently refined this result by showing that every such graph is a subgraph of some graph with treewidth [Formula: see text], where each vertex is blown up by a complete graph of order [Formula: see text]. Solving an open problem of Illingworth, Scott, and Wood [2022], we prove that the treewidth bound can be reduced to 4 while keeping blowups of order [Formula: see text]. As an extension of the Lipton–Tarjan theorem, in the case of planar graphs, we show that the treewidth can be further reduced to 2, which is best possible. We generalize this result for [Formula: see text]-minor-free graphs, with blowups of order [Formula: see text]. This setting includes graphs embeddable on any fixed surface.

Abstract. A circle graph is an intersection graph of a set of chords of a circle. We describe the unavoidable induced subgraphs of circle graphs with large treewidth. This includes examples that are far from the “usual suspects.” Our results imply that treewidth and Hadwiger number are linearly tied on the class of circle graphs and that the unavoidable induced subgraphs of a vertex-minor-closed class with large treewidth are the usual suspects if and only if the class has bounded rank-width. Using the same tools, we also study the treewidth of graphs [Formula: see text] that have a circular drawing whose crossing graph is well-behaved in some way. In this setting, we show that if the crossing graph is [Formula: see text]-minor-free, then [Formula: see text] has treewidth at most [Formula: see text] and has no [Formula: see text]-topological minor. On the other hand, we show that there are graphs with arbitrarily large Hadwiger number that have circular drawings whose crossing graphs are 2-degenerate.

Abstract. The planar graph product structure theorem of Dujmović et al. [ J. ACM, 67 (2020), 22] states that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. This result has been the key tool to resolve important open problems regarding queue layouts, nonrepetitive colorings, centered colorings, and adjacency labeling schemes. In this paper, we extend this line of research by utilizing shallow minors to prove analogous product structure theorems for several beyond-planar graph classes. The key observation that drives our work is that many beyond-planar graphs can be described as a shallow minor of the strong product of a planar graph with a small complete graph. In particular, we show that powers of bounded degree planar graphs, [Formula: see text]-planar, [Formula: see text]-cluster planar, fan-planar, and [Formula: see text]-fan-bundle planar graphs have such a shallow-minor structure. Using a combination of old and new results, we deduce that these classes have bounded queue-number, bounded nonrepetitive chromatic number, polynomial [Formula: see text]-centered chromatic numbers, linear strong coloring numbers, and cubic weak coloring numbers. In addition, we show that [Formula: see text]-gap planar graphs have at least exponential local treewidth and, as a consequence, cannot be described as a subgraph of the strong product of a graph with bounded treewidth and a path.