Abstract not found

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 >

We prove that planar graphs have O(log 4 n) queue number, thus improving upon the previous O ( √ n) upper bound. Consequently, planar graphs admit 3D straight-line crossingfree grid drawings in O(n log c n) volume, for some constant c, thus improving upon the previous O(n 3/2) upper bound. 2 1