## Tree-partitions of k-trees with applications in graph layout (2002)

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Venue: | Proc. 29th Workshop on Graph Theoretic Concepts in Computer Science (WG’03 |

Citations: | 15 - 11 self |

### BibTeX

@INPROCEEDINGS{Dujmović02tree-partitionsof,

author = {Vida Dujmović and David R. Wood},

title = {Tree-partitions of k-trees with applications in graph layout},

booktitle = {Proc. 29th Workshop on Graph Theoretic Concepts in Computer Science (WG’03},

year = {2002},

pages = {205--217},

publisher = {Springer}

}

### Years of Citing Articles

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### Abstract

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. 1

### Citations

515 | A heuristic for graph drawing
- Eades
- 1984
(Show Context)
Citation Context ...rawings. Motivated by applications in information visualisation, VLSI layout, and software engineering (see [12]), there is a growing body of research in three-dimensional straight-line graph drawing =-=[4, 6, 8, 12, 13, 24, 26, 35]-=-. The remainder of the paper is organised as follows. Section 1.1 recalls a number of definitions and well-known results. In Sections 1.2, 1.3 and 1.4 we survey and state our results for tree-partitio... |

294 |
Algorithmic aspects of vertex elimination on graphs
- Rose, Tarjan, et al.
- 1976
(Show Context)
Citation Context ...orm a clique of G, for all d ≥ 1. 1 In the language of chordal graphs, σ is a (reverse) perfect elimination vertex-ordering and can be determined, for example, by the Lex-BFS algorithm of Rose et al. =-=[18]-=-.sTree-Partitions of k-Trees with Applications in Graph Layout 209 Proof. We proceed by induction on i. The result is trivially true for i = 1. Suppose it is true for i − 1. Let d be the depth of vi. ... |

252 |
A decomposition theorem for partially ordered sets
- Dilworth
- 1950
(Show Context)
Citation Context ...3 v 4 v 5 w 1 w 2 w 3 w 4 w 5 Figure 1: A rainbow of size 5 in a vertex-ordering. A vertex-ordering containing a k-rainbow needs at least k queues. A straightforward application of Dilworth's Theorem =-=[9]-=- proves the converse. That is, a fixed vertex-ordering admits a k-queue layout where k is the size of the largest rainbow. (Heath and Rosenberg [23] describe an O(m log log n) time algorithm to comput... |

248 | A Partial k-Arboretum of Graphs with Bounded Treewidth
- Bodlaender
- 1998
(Show Context)
Citation Context ...r sub-class of the k-trees. A subgraph of a k-tree is called a partial k-tree, and a subgraph of a strict k-tree is called a partial strict k-tree. The following result is well known (see for example =-=[2, 27-=-]). Lemma 1. Let G be a graph. The following are equivalent: (1) G has tree-width tw(G) k, (2) G is a partial k-tree, (3) G is a partial strict k-tree, (4) G is a subgraph of a chordal graph with no ... |

152 |
Linear time algorithms for NP-hard problems restricted to partial k-trees
- Arnborg, Proskurowski
- 1989
(Show Context)
Citation Context ...is also a k-tree. This definition of a k-tree is by Rautenbach and Reed [27]. The following more restrictive definition of a k-tree, which we call `strict', was introduced by Arnborg and Proskurowski =-=[1]-=- and is more often used in the literature. A k-clique is a strict k-tree, and the graph obtained from a strict k-tree by adding a new vertex v adjacent to each vertex of a k-clique is also a strict k-... |

151 |
How to draw a planar graph on a grid
- Fraysseix, Pach, et al.
- 1990
(Show Context)
Citation Context ... open problem due to Felsner et al. [13], who ask whether every planar graph has a three-dimensional drawing with O(n) volume? A celebrated result independently due to de Fraysseix, Pach, and Pollack =-=[7]-=- and Schnyder [32] states that every planar graph has a two-dimensional straight-line grid drawing with O(n 2 ) area, and that n 2 ) area is necessary for certain planar graphs. Even whether every pla... |

48 | Embedding graphs in books: a layout problem with applications to VLSI design
- Chung, Leighton, et al.
- 1987
(Show Context)
Citation Context ...eue-number qn(G) of a graph G is the minimum, taken over all vertex-orderings of G, of the maximum size of a rainbow in . Stack and/or queue layouts of k-trees have previously been investigated in [5=-=, 15, 28, 35]-=-. A 1-tree has queue-number at most one, since in a lexicographical breadth-first vertex-ordering of a tree no two edges are nested. Chung et al. [5] proved that in a depth-first vertex-ordering of a ... |

43 |
Laying out graphs using queues
- HEATH, ROSENBERG
- 1992
(Show Context)
Citation Context ... two models of graph layout. The first, called a queue layout, consists of a total order of the vertices, and a partition of the edges into queues, such that no two edges in the same queue are nested =-=[11,12,15,17,20]-=-. The dual concept of a stack layout (or book embedding), is defined similarly, except that no two edges in the same stack may cross. The minimum number of queues (respectively, stacks) in a queue (st... |

38 | Straight-line drawings on restricted integer grids in two and three dimensions
- Felsner, Liotta, et al.
(Show Context)
Citation Context ...or queue-number. This result has significant implications for other open problems in the field. The second model of graph layout considered is that of a three-dimensional (straight-line grid) drawing =-=[2,3,5,8,14,20]-=-. Here vertices are positioned at gridpoints in Z 3 , and edges are drawn as straight line-segments with no crossings. ⋆ Research supported by NSERC and FCAR. H.L. Bodlaender (Ed.): WG 2003, LNCS 2880... |

33 |
Comparing queues and stacks as mechanisms for laying out graphs
- HEATH, LEIGHTON, et al.
- 1992
(Show Context)
Citation Context ... two models of graph layout. The first, called a queue layout, consists of a total order of the vertices, and a partition of the edges into queues, such that no two edges in the same queue are nested =-=[11,12,15,17,20]-=-. The dual concept of a stack layout (or book embedding), is defined similarly, except that no two edges in the same stack may cross. The minimum number of queues (respectively, stacks) in a queue (st... |

26 |
Tree-Partite Graphs and the Complexity of Algorithms
- Seese
- 1985
(Show Context)
Citation Context ...e-partition. The minimum width over all tree-partitions of a graph G is the tree-partition-width of G, denoted by tpw(G). Ding and Oporowski [4] proved that tpw(G) ≤ 24 tw(G) · max{1,∆(G)}, and Seese =-=[19]-=- proved that tw(G) ≤ 2 tpw(G) − 1, for every graph G. Theorem 1 below provides a tree-partition of a k-tree with additional features besides small width (see Figure 1). First, the subgraph induced by ... |

23 |
Some Results on Tree Decomposition of Graphs
- Ding, Oporowski
- 1995
(Show Context)
Citation Context ... 2 T x and w 2 T y (vw is called an inter-bag edge). The main property of tree-partitions which has been studied in the literature is the maximum size of a bag, called the width of the tree-partition =-=[3, 10, 11, 18, 33]-=-. The minimum width over all tree-partitions of a graph G is the tree-partition-width 1 of G, denoted by tpw(G). A graph with bounded degree has bounded tree-partition-width if and only if it has boun... |

23 | Path-width and threedimensional straight-line grid drawings of graphs
- Dujmović, Morin, et al.
- 2002
(Show Context)
Citation Context ...rid drawings with linear volume, which is the largest known class of graphs admitting such drawings. Motivated by applications in information visualisation, VLSI layout, and software engineering (see =-=[12]-=-), there is a growing body of research in three-dimensional straight-line graph drawing [4, 6, 8, 12, 13, 24, 26, 35]. The remainder of the paper is organised as follows. Section 1.1 recalls a number ... |

20 | Three Dimensional Grid Drawings of Graphs
- Pach, Thiele, et al.
- 1999
(Show Context)
Citation Context ...or queue-number. This result has significant implications for other open problems in the field. The second model of graph layout considered is that of a three-dimensional (straight-line grid) drawing =-=[2,3,5,8,14,20]-=-. Here vertices are positioned at gridpoints in Z 3 , and edges are drawn as straight line-segments with no crossings. ⋆ Research supported by NSERC and FCAR. H.L. Bodlaender (Ed.): WG 2003, LNCS 2880... |

18 | Three-dimensional grid drawings with subquadratic volume - Dujmović, Wood - 2004 |

17 |
Algorithmic aspects of tree width
- REED
- 2003
(Show Context)
Citation Context ...s follows. The empty graph is a k-tree, and the graph obtained from a k-tree by adding a new vertex adjacent to each vertex of a clique with at most k vertices is a k-tree. This definition is by Reed =-=[16]-=-. The following more common definition of a k-tree, which we call ‘strict’, was introduced by Arnborg and Proskurowski [1]. A k-clique is a strict k-tree, and the graph obtained from a strict k-tree b... |

16 |
3D straight-line grid drawing of 4colorable graphs
- Calamoneri, Sterbini
- 1997
(Show Context)
Citation Context ...rawings. Motivated by applications in information visualisation, VLSI layout, and software engineering (see [12]), there is a growing body of research in three-dimensional straight-line graph drawing =-=[4, 6, 8, 12, 13, 24, 26, 35]-=-. The remainder of the paper is organised as follows. Section 1.1 recalls a number of definitions and well-known results. In Sections 1.2, 1.3 and 1.4 we survey and state our results for tree-partitio... |

16 | Drawing series-parallel graphs on a box
- Giacomo, Liotta, et al.
(Show Context)
Citation Context ...rawings. Motivated by applications in information visualisation, VLSI layout, and software engineering (see [12]), there is a growing body of research in three-dimensional straight-line graph drawing =-=[4, 6, 8, 12, 13, 24, 26, 35]-=-. The remainder of the paper is organised as follows. Section 1.1 recalls a number of definitions and well-known results. In Sections 1.2, 1.3 and 1.4 we survey and state our results for tree-partitio... |

14 |
Exploring the Powers of Stacks and Queues via Graph Layouts
- PEMMARAJU
- 1992
(Show Context)
Citation Context ... two models of graph layout. The first, called a queue layout, consists of a total order of the vertices, and a partition of the edges into queues, such that no two edges in the same queue are nested =-=[11,12,15,17,20]-=-. The dual concept of a stack layout (or book embedding), is defined similarly, except that no two edges in the same stack may cross. The minimum number of queues (respectively, stacks) in a queue (st... |

12 | Queue layouts, tree-width, and three-dimensional graph drawing
- Wood
(Show Context)
Citation Context |

11 | On tree-partitions of graphs
- Ding, Oporowski
- 1996
(Show Context)
Citation Context ... 2 T x and w 2 T y (vw is called an inter-bag edge). The main property of tree-partitions which has been studied in the literature is the maximum size of a bag, called the width of the tree-partition =-=[3, 10, 11, 18, 33]-=-. The minimum width over all tree-partitions of a graph G is the tree-partition-width 1 of G, denoted by tpw(G). A graph with bounded degree has bounded tree-partition-width if and only if it has boun... |

11 |
Three-dimensional graph drawing, Algorithmica 17
- Cohen, Eades, et al.
- 1997
(Show Context)
Citation Context ...or queue-number. This result has significant implications for other open problems in the field. The second model of graph layout considered is that of a three-dimensional (straight-line grid) drawing =-=[2,3,5,8,14,20]-=-. Here vertices are positioned at gridpoints in Z 3 , and edges are drawn as straight line-segments with no crossings. ⋆ Research supported by NSERC and FCAR. H.L. Bodlaender (Ed.): WG 2003, LNCS 2880... |

11 |
The pagenumber of k-trees is
- Ganley, Heath
(Show Context)
Citation Context ...t-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 fie... |

9 | Multitragk Interval Graphs
- Gy-fis, West
- 1995
(Show Context)
Citation Context ...d layering with no X-crossing and no intralayer edges’ in [5,6,20]. Similar structures are implicit in [8,11,12,17]. Note that this definition of track-number is unrelated to that of Gyárfás and West =-=[10]-=-.sTree-Partitions of k-Trees with Applications in Graph Layout 211 Lemma 2. [6] Let L ⊆ I be a set of tracks in a track layout {(Vi,<i) :i ∈ I} of a graph G. IfS is a set of cliques, each of which cov... |

9 |
Veni Madhavan, Stack and queue number of 2-trees
- Rengarajan, E
- 1995
(Show Context)
Citation Context |

3 |
On tree-partitions of graphs, Discrete Math
- Ding, Oporowski
- 1996
(Show Context)
Citation Context ... v ∈ Tx and w ∈ Ty (vw is called an inter-bag edge). The main property of tree-partitions which has been studied in the literature is the maximum size of a bag, called the width of the tree-partition =-=[3, 10, 11, 18, 33]-=-. The minimum width over all tree-partitions of a graph G is the tree-partition-width 1 of G, denoted by tpw(G). A graph with bounded degree has bounded tree-partition-width if and only if it has boun... |

1 |
Drawing series-parallel graphs on abox
- Giacomo, Liotta, et al.
- 2002
(Show Context)
Citation Context |

1 |
Pagenumber and treewidth
- Lin, Li
(Show Context)
Citation Context ...s 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]-=-. A 1-tree has queue-number at most one, since in a lexicographical breadthfirst vertex-ordering of a tree no two edges are nested [12]. Rengarajan and Veni Madhavan [17] proved that 2-trees have queu... |