Results 1  10
of
25
On Linear Layouts of Graphs
, 2004
"... In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp... ..."
Abstract

Cited by 30 (18 self)
 Add to MetaCart
In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp...
Planar decompositions and the crossing number of graphs with an excluded minor
 IN GRAPH DRAWING 2006; LECTURE NOTES IN COMPUTER SCIENCE 4372
, 2007
"... 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 ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
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 treewidth has linear convex crossing number, and every K3,3minorfree graph with bounded degree has linear rectilinear crossing number.
Drawing Kn in Three Dimensions with One Bend per Edge
, 2006
"... We give a drawing of Kn in three dimensions in which vertices are placed at integer grid points and edges are drawn crossingfree with at most one bend per edge in a volume bounded by O(n^2.5). ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
We give a drawing of Kn in three dimensions in which vertices are placed at integer grid points and edges are drawn crossingfree with at most one bend per edge in a volume bounded by O(n^2.5).
A note on treepartitionwidth
, 2006
"... Abstract. A treepartition of a graph G is a proper partition of its vertex set into ‘bags’, such that identifying the vertices in each bag produces a forest. The treepartitionwidth of G is the minimum number of vertices in a bag in a treepartition of G. An anonymous referee of the paper by Ding ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
Abstract. A treepartition of a graph G is a proper partition of its vertex set into ‘bags’, such that identifying the vertices in each bag produces a forest. The treepartitionwidth of G is the minimum number of vertices in a bag in a treepartition of G. An anonymous referee of the paper by Ding and Oporowski [J. Graph Theory, 1995] proved that every graph with treewidth k ≥ 3 and maximum degree ∆ ≥ 1 has treepartitionwidth at most 24k∆. We prove that this bound is within a constant factor of optimal. In particular, for all k ≥ 3 and for all sufficiently large ∆, we construct a graph with treewidth k, maximum degree ∆, and treepartitionwidth at least ( 1 8 upper bound to 5
CHARACTERISATIONS AND EXAMPLES OF GRAPH CLASSES WITH BOUNDED EXPANSION
"... 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 t ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
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 lineartime 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 socalled 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 ErdösRényi 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 nonrepetitive chromatic number. We also prove that graphs with ‘linear ’ crossing number are contained in a topologicallyclosed class, while graphs with bounded crossing number are contained in a minorclosed class.
Notes on nonrepetitive graph colouring
 Electron. J. Combin
, 2008
"... 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 kcolouring that is nonrepetitive on paths. We prove that every graph has a sub ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
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 kcolouring that is nonrepetitive on paths. We prove that every graph has a subdivision that admits a 4colouring 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
ThreeDimensional 1Bend Graph Drawings
 Concordia University
, 2004
"... We consider threedimensional griddrawings 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 ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
We consider threedimensional griddrawings 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 threedimensional griddrawing with O(n 3 / log 2 n) volume and one bend per edge. The best previous bound was O(n 3).
On the Queue Number of Planar Graphs
, 2010
"... 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 straightline 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 ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
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 straightline 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
Layouts of Graph Subdivisions
, 2004
"... A kstack layout (respectively, kqueue layout) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of noncrossing (nonnested) edges with respect to the vertex ordering. A ktrack layout of a graph consists of a vertex kcolouring, and a total order of e ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
A kstack layout (respectively, kqueue layout) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of noncrossing (nonnested) edges with respect to the vertex ordering. A ktrack layout of a graph consists of a vertex kcolouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. The stacknumber (respectively, queuenumber, tracknumber) of a graph G, denoted by sn(G) (qn(G), tn(G)), is the minimum k such that G has a kstack (kqueue, ktrack) layout. This paper studies stack, queue, and track layouts of graph subdivisions. It is known that every graph has a 3stack subdivision. The best known upper bound on the number of division vertices per edge in a 3stack subdivision of an nvertex graph G is improved from O(log n) to O(log min{sn(G), qn(G)}). This result reduces the question of whether queuenumber is bounded by stacknumber to whether 3stack graphs have bounded queue number. It is proved that every graph has a 2queue subdivision, a 4track subdivision, and a mixed 1stack 1queue subdivision. All these values are optimal for every nonplanar graph. In addition, we characterise those graphs with kstack, kqueue, and ktrack subdivisions, for all values of k. The number of division vertices per edge in the case of 2queue and 4track subdivisions, namely O(log qn(G)), is optimal to within a constant factor, for every graph G. Applications to 3D polyline grid drawings are presented. For example, it is proved that every graph G has a 3D polyline grid drawing with the vertices on a rectangular prism, and with O(log qn(G)) bends per edge.
Characterizations and Examples of Graph Classes with bounded expansion
"... 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 th ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
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 lineartime 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 socalled 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 ErdösRényi model of random graphs with constant average degree. In particular, we prove that for every fixed d >