Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an ℓ × ℓ grid minor is exponential in ℓ. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A grid-like-minor of order ℓ in a graph G is a set of paths in G whose intersection graph is bipartite and contains a Kℓ-minor. For example, the rows and columns of the ℓ × ℓ grid are a grid-like-minor of order ℓ + 1. We prove that polynomial treewidth forces a large grid-like-minor. In particular, every graph with treewidth at least cℓ 4 √ log ℓ has a grid-like-minor of order ℓ. As an application of this result, we prove that the cartesian product G □ K2 contains a Kℓ-minor whenever G has treewidth at least cℓ 4 √ log ℓ.

Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Neˇsetˇril and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of the depth of the shallow minor. Several linear-time algorithms are known for bounded expansion classes (such as subgraph isomorphism testing), and they allow restricted homomorphism dualities, amongst other desirable properties. In this paper we establish two new characterisations of bounded expansion classes, one in terms of so-called topological parameters, the other in terms of controlling dense parts. The latter characterisation is then used to show that the notion of bounded expansion is compatible with Erdös-Rényi model of random graphs with constant average degree. In particular, we prove that for every fixed d> 0, there exists a class with bounded expansion, such that a random graph of order n and edge probability d/n asymptotically almost surely belongs to the class. We then present several new examples of classes with bounded expansion that do not exclude some topological minor, and appear naturally in the context of graph drawing or graph colouring. In particular, we prove that the following classes have bounded expansion: graphs that can be drawn in the plane with a bounded number of crossings per edge, graphs with bounded stack number, graphs with bounded queue number, and graphs with bounded non-repetitive chromatic number. We also prove that graphs with ‘linear ’ crossing number are contained in a topologically-closed class, while graphs with bounded crossing number are contained in a minor-closed class.

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Graph separators are a ubiquitous tool in graph theory and computer science. However, in some applications, their usefulness is limited by the fact that the separator can be as large as Ω ( √ n) in graphs with n vertices. This is the case for planar graphs, and more generally, for proper minor-closed families. We study a special type of graph separator, called a layered separator, which possibly has linear size in n, but has constant size with respect to a different measure, called the breadth. We prove that a wide class of graphs admit layered separators of bounded breadth, including graphs of bounded Euler genus. We use these results to prove O(log n) bounds for a number of problems where O ( √ n) was a long standing previous best bound. This includes queue-number and nonrepetitive chromatic number of bounded Euler genus graphs. We extend these results, with a log O(1) n bound, to all proper minor-closed families. This result also implies that every graph from a proper minor-closed class has a 3-dimensional grid drawing in n log O(1) n volume, where the previous best bound was O(n 3/2). Only for planar graphs was a log O(1) n bound on the queue-number previously known.

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Abstract: The following theorem is proved: for all k-connected graphs G and H each with at least n vertices, the treewidth of the cartesian product of G and H is at least k(n − 2k + 2) − 1. For n k, this lower bound is asymptotically tight for particular graphsG andH. This theorem generalizes a well-known result about the treewidth of planar grid graphs. C © 2012 Wiley