Upward Three-Dimensional Grid Drawings of Graphs

Dujmović, Vida; Wood, David R.

doi:10.1007/s11083-006-9028-y

Cited by 15 publications

(9 citation statements)

References 44 publications

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“…Dujmović and Wood [31] proved that every graph with bounded maximum degree has a 3-dimensional grid drawing with volume O(n 3/2 ), and the same bound holds for graphs from a proper minor-closed class. In fact, every graph with bounded degeneracy has a 3-dimensional grid drawing with O(n 3/2 ) volume [33]. Dujmović et al [27] proved that every graph with bounded treewidth has a 3-dimensional grid drawing with volume O(n).…”

Section: Queue-number and 3-dimensional Grid Drawingsmentioning

confidence: 99%

Layered separators in minor-closed graph classes with applications

Dujmović

Morin

Wood

2017

Journal of Combinatorial Theory, Series B

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110

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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 classes. We study a special type of graph separator, called a layered separator, which may have linear size in n, but has bounded size with respect to a different measure, called the width. We prove, for example, that planar graphs and graphs of bounded Euler genus admit layered separators of bounded width. More generally, we characterise the minor-closed classes that admit layered separators of bounded width as those that exclude a fixed apex graph as a minor.We use layered separators to prove O(log n) bounds for a number of problems where O( √ n) was a long-standing previous best bound. This includes the nonrepetitive chromatic number and queue-number of graphs with bounded Euler genus. We extend these results with a O(log n) bound on the nonrepetitive chromatic number of graphs excluding a fixed topological minor, and a log O(1) n bound on the queue-number of graphs excluding a fixed minor. Only for planar graphs were log O(1) n bounds previously known. Our results imply that every n-vertex graph excluding a fixed minor has a 3-dimensional grid drawing with n log O(1) n volume, whereas the previous best bound was O(n 3/2 ).

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Section: Queue-number and 3-dimensional Grid Drawingsmentioning

confidence: 99%

Layered separators in minor-closed graph classes with applications

Dujmović

Morin

Wood

2017

Journal of Combinatorial Theory, Series B

Self Cite

110

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“…We conclude this section with a final observation about repetitive colourings. Dujmović and Wood [29] defined a strong star colouring of a graph G to be a colouring such that between each pair of colour classes all the edges are incident to a single vertex. That is, each bichromatic subgraph is a star plus isolated vertices.…”

Section: Lemma 92 ([70]mentioning

confidence: 99%

Characterisations and examples of graph classes with bounded expansion

Nešetřil

Mendez

Wood

2012

European Journal of Combinatorics

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103

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Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Nešetřil 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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“…In particular, this implies that graphs with three-dimensional drawings of linear volume have only linear many edges. Dujmović and Wood [12] showed that graphs with bounded degeneracy admit drawings with O(n 3/2 ) volume. A major open problem in this area is due to Felsner et al [14] and asks whether planar graphs can be drawn with linear volume.…”

Section: Introductionmentioning

confidence: 99%

On the Queue-Number of Graphs with Bounded Tree-Width

Wiechert

2017

Electron. J. Combin.

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A queue layout of a graph consists of a linear order on the vertices and an assignment of the edges to queues, such that no two edges in a single queue are nested. The minimum number of queues needed in a queue layout of a graph is called its queue-number.We show that for each k 1, graphs with tree-width at most k have queuenumber at most 2 k − 1. This improves upon double exponential upper bounds due to Dujmović et al. and Giacomo et al. As a consequence we obtain that these graphs have track-number at most 2 O(k 2 ) .We complement these results by a construction of k-trees that have queuenumber at least k + 1. Already in the case k = 2 this is an improvement to existing results and solves a problem of Rengarajan and Veni Madhavan, namely, that the maximal queue-number of 2-trees is equal to 3.

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Upward Three-Dimensional Grid Drawings of Graphs

Cited by 15 publications

References 44 publications

Layered separators in minor-closed graph classes with applications

Layered separators in minor-closed graph classes with applications

Characterisations and examples of graph classes with bounded expansion

On the Queue-Number of Graphs with Bounded Tree-Width

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