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

Wiechert, Veit

doi:10.37236/6429

Cited by 25 publications

(34 citation statements)

References 25 publications

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“…Wood [76] established that qn(G ⊠ H) 2 sqn(H) • qn(G) + sqn(H) + qn(G) for all graphs G and H. Dujmović, Morin, and Wood [28] proved that graphs of bounded treewidth have bounded queue-number. The best known bound is due to Wiechert [75] who showed that qn(G) 2 tw(G) − 1. Using similar techniques to the result of Wood [76], Dujmović et al [27] proved the following.…”

Section: Stack and Queue Layoutsmentioning

confidence: 99%

Shallow Minors, Graph Products and Beyond Planar Graphs

Hickingbotham¹,

Wood²

2021

Preprint

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The planar graph product structure theorem of Dujmović, Joret, Micek, Morin, Ueckerdt, and Wood [J. ACM 2020] states that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. This result has been the key tool to resolve important open problems regarding queue layouts, nonrepetitive colourings, centered colourings, and adjacency labelling schemes. In this paper, we extend this line of research by utilizing shallow minors to prove analogous product structure theorems for several beyond planar graph classes. The key observation that drives our work is that many beyond planar graphs can be described as a shallow minor of the strong product of a planar graph with a small complete graph. In particular, we show that power of planar graphs, k-planar, (k, p)-cluster planar, k-semi-fan-planar graphs and k-fan-bundle planar graphs can be described in this manner. Using a combination of old and new results, we deduce that these classes have bounded queuenumber, bounded nonrepetitive chromatic number, polynomial p-centred chromatic numbers, linear strong colouring numbers, and cubic weak colouring numbers. In addition, we show that k-gap planar graphs have super-linear local treewidth and, as a consequence, cannot be described as a subgraph of the strong product of a graph with bounded treewidth and a path.

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Section: Stack and Queue Layoutsmentioning

confidence: 99%

Shallow Minors, Graph Products and Beyond Planar Graphs

Hickingbotham¹,

Wood²

2021

Preprint

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“…Differences with Wiechert's algorithm. Wiechert's algorithm [23] builds upon a previous algorithm by Dujmović et al [6]. Both yield queue layouts for general k-trees, using the breadth-first search (BFS) starting from an arbitrary vertex r of G. For each d > 0 and each connected component C induced by the vertices at distance d from r, create a node (called bag) "containing" all vertices of C; two bags are adjacent if there is an edge of G between them.…”

Section: Lemmamentioning

confidence: 99%

“…In Section 2, we improve the upper bound on the queue number of planar 3-trees from seven [23] to five; recall that a planar 3-tree is a triangulated plane graph G with n ≥ 3 vertices, such that G is either a 3-cycle, if n = 3, or has a vertex whose deletion gives a planar 3-tree with n−1 vertices, if n > 3. In Section 3, we show that there exist planar 3-trees, whose queue number is at least four, thus strengthening a corresponding result of Wiechert [23] for general (that is, not necessarily planar) 3-trees. We stress that our lower bound is also the best known for planar graphs.…”

Section: Introductionmentioning

confidence: 99%

Queue Layouts of Planar 3-Trees

et al. 2020

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A queue layout of a graph G consists of a linear order of the vertices of G and a partition of the edges of G into queues, so that no two independent edges of the same queue are nested. The queue number of G is the minimum number of queues required by any queue layout of G.In this paper, we continue the study of the queue number of planar 3trees. As opposed to general planar graphs, whose queue number is not known to be bounded by a constant, the queue number of planar 3-trees has been shown to be at most seven. In this work, we improve the upper bound to five. We also show that there exist planar 3-trees, whose queue number is at least four; this is the first example of a planar graph with queue number greater than three.

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“…Their bound on the queue-number was doubly exponential in the treewidth. Wiechert [98] improved this bound to singly exponential.…”

Section: Treewidth and Layered Treewidthmentioning

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

Cited by 25 publications

References 25 publications

Shallow Minors, Graph Products and Beyond Planar Graphs

Shallow Minors, Graph Products and Beyond Planar Graphs

Queue Layouts of Planar 3-Trees

Planar Graphs have Bounded Queue-Number

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