In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A k-stack (resp...

Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar decomposition with bounded width and a linear number of bags. The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. We prove that planar decompositions are intimately related to the crossing number. In particular, a graph with bounded degree has linear crossing number if and only if it has a planar decomposition with bounded width and linear order. It follows from the above result about planar decompositions that every graph with bounded degree and an excluded minor has linear crossing number. Analogous results are proved for the convex and rectilinear crossing numbers. In particular, every graph with bounded degree and bounded tree-width has linear convex crossing number, and every K3,3-minor-free graph with bounded degree has linear rectilinear crossing number.

We give a drawing of Kn in three dimensions in which vertices are placed at integer grid points and edges are drawn crossing-free with at most one bend per edge in a volume bounded by O(n^2.5).

We consider three-dimensional grid-drawings of graphs with at most one bend per edge. Under the additional requirement that the vertices be collinear, we prove that the minimum volume of such a drawing is Θ(cn), where n is the number of vertices and c is the cutwidth of the graph. We then prove that every graph has a three-dimensional grid-drawing with O(n 3 / log 2 n) volume and one bend per edge. The best previous bound was O(n 3).

A vertex colouring of a graph is nonrepetitive on paths if there is no path v1, v2,..., v2t such that vi and vt+i receive the same colour for all i = 1, 2,..., t. We determine the maximum density of a graph that admits a k-colouring that is nonrepetitive on paths. We prove that every graph has a subdivision that admits a 4-colouring that is nonrepetitive on paths. The best previous bound was 5. We also study colourings that are nonrepetitive on walks, and provide a conjecture that would imply that every graph with maximum degree ∆ has a f(∆)-colouring that is nonrepetitive on walks. We prove that every graph with treewidth k and maximum degree ∆ has a O(k∆)-colouring that is nonrepetitive on paths, and a O(k ∆ 3)-colouring that is nonrepetitive on walks. 1

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.

Abstract. A k-tree is a chordal graph with no (k + 2)-clique. An ℓ-treepartition of a graph G is a vertex partition of G into ‘bags’, such that con-tracting each bag to a single vertex gives an ℓ-tree (after deleting loops and replacing parallel edges by a single edge). We prove that for all k ≥ ℓ ≥ 0, every k-tree has an ℓ-tree-partition in which every bag induces a connected ⌊k/(ℓ + 1)⌋-tree. An analogous result is proved for oriented k-trees. 1.

A d-dimensional hypercube drawing of a graph represents the vertices by distinct points in {0, 1} d, such that the line-segments representing the edges do not cross. We study lower and upper bounds on the minimum number of dimensions in hypercube drawing of a given graph. This parameter turns out to be related to Sidon sets and antimagic injections. 1

We study straight-line drawings of non-planar graphs with few slopes. Interval graphs, co-comparability graphs and AT-free graphs are shown to have have drawings in which the number of slopes is bounded by the maximum degree. We prove that graphs of bounded degree and bounded treewidth have drawings with O(log n) slopes. Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the maximum degree. In a companion paper, planar drawings of graphs with few slopes are also considered.

Abstract. A three-dimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line segments representing the edges do not cross. Our aim is to produce three-dimensional grid drawings with small bounding box volume. Our first main result is that every n-vertex graph with bounded degeneracy has a three-dimensional grid drawing with O(n 3/2) volume. This is the largest known class of graphs that have such drawings. A three-dimensional grid drawing of a directed acyclic graph (dag) is upward if every arc points up in the z-direction. We prove that every dag has an upward three-dimensional grid drawing with O(n 3) volume, which is tight for the complete dag. The previous best upper bound was O(n 4). Our main result concerning upward drawings is that every c-colourable dag (c constant) has an upward three-dimensional grid drawing with O(n 2) volume. This result matches the bound in the undirected case, and improves the best known bound from O(n 3) for many classes of dags, including planar, series parallel, and outerplanar. Improved bounds are also obtained for tree dags. We prove a strong relationship between upward three-dimensional grid drawings, upward track layouts, and upward queue layouts. Finally, we study upward three-dimensional grid drawings with bends in the edges. 1.