Track Layouts, Layered Path Decompositions, and Leveled Planarity

Bannister, Michael J.; Devanny, William E.; Dujmović, Vida; Eppstein, David; Wood, David R.

doi:10.1007/s00453-018-0487-5

Cited by 40 publications

(70 citation statements)

References 29 publications

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“…Applications of layered treewidth include graph colouring [46,70,77], graph drawing [10,46], book embeddings [44], and intersection graph theory [96]. The related notion of layered pathwidth has also been studied [10,41]. Most relevant to this paper, Dujmović et al [46] proved that every graph with n vertices and layered treewidth k has queue-number at most O(k log n).…”

Section: Lemma 4 ([98]mentioning

confidence: 99%

Planar Graphs have Bounded Queue-Number

Dujmović

Joret

Micek

et al. 2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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We show that planar graphs have bounded queue-number, thus proving a conjecture of Heath, Leighton and Rosenberg from 1992. The key to the proof is a new structural tool called layered partitions, and the result that every planar graph has a vertex-partition and a layering, such that each part has a bounded number of vertices in each layer, and the quotient graph has bounded treewidth. This result generalises for graphs of bounded Euler genus. Moreover, we prove that every graph in a minor-closed class has such a layered partition if and only if the class excludes some apex graph. Building on this work and using the graph minor structure theorem, we prove that every proper minor-closed class of graphs has bounded queue-number.Layered partitions have strong connections to other topics, including the following two examples. First, they can be interpreted in terms of strong products. We show that every planar graph is a subgraph of the strong product of a path with some graph of bounded treewidth. Similar statements hold for all proper minor-closed classes. Second, we give a simple proof of the result by DeVos et al. (2004) that graphs in a proper minor-closed class have low treewidth colourings.

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Section: Lemma 4 ([98]mentioning

confidence: 99%

Planar Graphs have Bounded Queue-Number

Dujmović

Joret

Micek

et al. 2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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“…We will use the following corollary of the theory: Corollary 1 Let P = (X, <) be a partial order on n elements, then there is an orthogonal pair A, C where A is a k-antichain and C an -chain and k + Fig. 4 together with two orthogonal pairs of L corresponding to the boundary points (1,3) and (3,1) of G(L); chains of C are blue, antichains of A are red, green, and yellow.…”

Section: Orthogonal Partitions Of Posetsmentioning

confidence: 99%

“…Clearly π(G) = 1 if and only if G is a forest of paths. The set of graphs with π(G) = 2, however, is already surprisingly rich, it contains trees, outerplanar graphs and subgraphs of grids, see [1,8].…”

Section: Introductionmentioning

confidence: 99%

4-Connected Triangulations on Few Lines

Felsner

2019

Lecture Notes in Computer Science

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We show that 4-connected plane triangulations can be redrawn such that edges are represented by straight segments and the vertices are covered by a set of at most p 2n lines each of them horizontal or vertical. The same holds for all subgraphs of such triangulations.The proof is based on a corresponding result for diagrams of planar lattices which makes use of orthogonal chain and antichain families.

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“…While it is easy to decide efficiently whether a graph admits a 2-track layout, Bannister et al [1] answered the open question of Dujmović et al already for 3-track layouts in the affirmative. They first showed that a graph has a leveled planar drawing if and only if it is bipartite and has a 3-track layout.…”

Section: Introductionmentioning

confidence: 99%

“…Combining this results with the NP-hardness of level planarity, proven by Heath and Rosenberg [13], immediately showed that it is NP-hard to decide whether a given graph has a 3-track layout. For k > 3, deciding the existence of a k-track layout is NP-hard, too, since it suffices to add to the given graph k − 3 new vertices each of which is incident to all original vertices of the graph [1].…”

Section: Introductionmentioning

confidence: 99%

Line and Plane Cover Numbers Revisited

Biedl

Felsner

Meijer

et al. 2019

Lecture Notes in Computer Science

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A measure for the visual complexity of a straight-line crossingfree drawing of a graph is the minimum number of lines needed to cover all vertices. For a given graph G, the minimum such number (over all drawings in dimension d ∈ {2, 3}) is called the d-dimensional weak line cover number and denoted by π 1 d (G). In 3D, the minimum number of planes needed to cover all vertices of G is denoted by π 2 3 (G). When edges are also required to be covered, the corresponding numbers ρ 1 d (G) and ρ 2 3 (G) are called the (strong) line cover number and the (strong) plane cover number. Computing any of these cover numbers-except π 1 2 (G)-is known to be NP-hard. The complexity of computing π 1 2 (G) was posed as an open problem by Chaplick et al. [WADS 2017]. We show that it is NP-hard to decide, for a given planar graph G, whether π 1 2 (G) = 2. We further show that the universal stacked triangulation of depth d, G d , has π 1 2 (G d ) = d+1. Concerning 3D, we show that any n-vertex graph G with ρ 2 3 (G) = 2 has at most 5n − 19 edges, which is tight.

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Track Layouts, Layered Path Decompositions, and Leveled Planarity

Cited by 40 publications

References 29 publications

Planar Graphs have Bounded Queue-Number

Planar Graphs have Bounded Queue-Number

4-Connected Triangulations on Few Lines

Line and Plane Cover Numbers Revisited

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