The pagenumber of k-trees is O(k)

Ganley, Joseph L.; Heath, Lenwood S.

doi:10.1016/s0166-218x(00)00178-5

Cited by 47 publications

(23 citation statements)

References 11 publications

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“…The minimum number of queues (respectively, stacks) in a queue (stack) layout of G is called the queue-number (stack-number or page-number ) of G, and is denoted by qn(G) (sn(G)). Ganley and Heath [9] proved that stack-number is bounded by tree-width, and asked whether queue-number is also bounded by tree-width? The bound of sn(G) ≤ tw(G) + 1 by Ganley and Heath [9] has recently been improved to sn(G) ≤ tw(G) by Lin and Li [13].…”

Section: Queue Layouts and 3d Graph Drawingsmentioning

confidence: 99%

“…Heath et al [11] conjectured that both of these questions have an affirmative answer. More recently however, Pemmaraju [15] conjectured that the 'stellated K 3 ', a planar 3-tree, has Θ(log n) queue-number, and provided evidence to support this conjecture (also see [9]). This suggested that the answers to the above questions were both negative.…”

Section: Open Problemsmentioning

confidence: 99%

“…Applications of queue layouts include parallel process scheduling, fault-tolerant processing, matrix computations, and sorting networks (see [15]). We prove that graphs of bounded tree-width have bounded queue-number, thus solving an open problem due to Ganley and Heath [9], who proved that stack-number is bounded by tree-width, and asked whether an analogous relationship holds for queue-number. This result has significant implications for other open problems in the field.…”

Section: Introductionmentioning

confidence: 97%

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Tree-Partitions of k-Trees with Applications in Graph Layout

Dujmović

Wood

2003

Graph-Theoretic Concepts in Computer Science

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Abstract.A tree-partition of a graph is a partition of its vertices into 'bags' such that contracting each bag into a single vertex gives a forest. It is proved that every k-tree has a tree-partition such that each bag induces a (k − 1)-tree, amongst other properties. Applications of this result to two well-studied models of graph layout are presented. First it is proved that graphs of bounded tree-width have bounded queuenumber, thus resolving an open problem due to Ganley and Heath [2001] and disproving a conjecture of Pemmaraju [1992]. This result provides renewed hope for the positive resolution of a number of open problems regarding queue layouts. In a related result, it is proved that graphs of bounded tree-width have three-dimensional straight-line grid drawings with linear volume, which represents the largest known class of graphs with such drawings.

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Section: Queue Layouts and 3d Graph Drawingsmentioning

confidence: 99%

Section: Open Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

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Tree-Partitions of k-Trees with Applications in Graph Layout

Dujmović

Wood

2003

Graph-Theoretic Concepts in Computer Science

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“…-Stacknumber: complete graphs [4], complete bipartite graphs [24], butterfly graphs [12], trees, grids, X-trees [5], hypercubes [5,22], de Bruijn digraphs, Kautz digraphs, shuffle-exchange graphs [16], planar graphs [32], genus-g graphs [23], bandwidth-k graphs [28], k-trees [13], iterated line digraphs [14]. -Queuenumber: complete graphs, complete bipartite graphs, trees, grids, unicyclic graphs, X-trees, binary de Bruijn graphs, butterfly graphs (all in [21]), k-tree [6,26,31],…”

Section: (A) ≤ σ(B) σ(C) ≤ σ(D) and σ(A) ≤ σ(C) Then One Of The Fomentioning

confidence: 99%

Laying Out Iterated Line Digraphs Using Queues

Hasunuma

2004

Graph Drawing

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Abstract. In this paper, we study a layout problem of a digraph using queues. The queuenumber of a digraph is the minimum number of queues required for a queue layout of the digraph. We present upper and lower bounds on the queuenumber of an iterated line digraph L k (G) of a digraph G. In particular, our upper bound depends only on G and is independent of the number of iterations k. Queue layouts can be applied to three-dimensional drawings. From the result on the queuenumber of L k (G), it is shown that for any fixed digraph G, L k (G) has a three-dimensional drawing with O(n) volume, where n is the number of vertices in L k (G). We also apply these results to particular families of iterated line digraphs such as de Bruijn digraphs, Kautz digraphs, butterfly digraphs, and wrapped butterfly digraphs.

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“…Also, Wood [20] shows that book-embeddings can be applied to three-dimensional graph drawings. Until now, book-embeddings have been studied for many graph classes: complete graphs [1], complete bipartite graphs [16], butterfly networks [6], trees, grids, X-trees [2], hypercubes [2,13], incomplete hypercubes [5], supercubes [4], de Bruijn graphs, Kautz graphs, shuffle-exchange graphs [9], planar graphs [21], genus-g graphs [15], bandwidth-k graphs [18], ktrees [7] and iterated line digraphs [8].…”

Section: Introductionmentioning

confidence: 99%

Improved book-embeddings of incomplete hypercubes

Hasunuma

2009

Discrete Applied Mathematics

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a b s t r a c tIn this paper, we show that any incomplete hypercube with, at most, 2 n + 2 n−1 + 2 n−2 vertices can be embedded in n − 1 pages for all n ≥ 4. For the case n ≥ 4, this result improves Fang and Lai's result that any incomplete hypercube with, at most, 2 n + 2 n−1 vertices can be embedded in n − 1 pages for all n ≥ 2.Besides this, we show that the result can be further improved when n is largee.g., any incomplete hypercube with at most 2 n + 2 n−1 + 2 n−2 + 2 n−7 (respectively, 2 n + 2 n−1 + 2 n−2 + 2 n−7 + 2 n−230 ) vertices can be embedded in n − 1 pages for all n ≥ 9 (respectively, n ≥ 232).

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The pagenumber of k-trees is O(k)

Cited by 47 publications

References 11 publications

Tree-Partitions of k-Trees with Applications in Graph Layout

Tree-Partitions of k-Trees with Applications in Graph Layout

Laying Out Iterated Line Digraphs Using Queues

Improved book-embeddings of incomplete hypercubes

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