## On Linear Layouts of Graphs (2004)

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### BibTeX

@MISC{Dujmovic04onlinear,

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

title = {On Linear Layouts of Graphs},

year = {2004}

}

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### Abstract

In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A k-stack (resp...

### Citations

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Reducibility among Combinatorial Problems. In
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Citation Context ...h comprised of k components each isomorphic to G. We claim that χ(G) ≤ k if and only if G′ is almost k-colourable. The result will follow from Theorem 1 and since graph k-colourability is NP-complete =-=[64]-=-. If G is k-colourable then so is G′, and thus G′ is almost k-colourable. Conversely, if G′ is almost kcolourable then there is a set S of at most k − 1 vertices such that χ(G′ \ S) ≤ k. Since |S| ≤ k... |

1195 | Algorithmic Graph Theory And Perfect Graphs - Golumbic - 1980 |

926 | Parameterized Complexity
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Citation Context ...een pointed out in Garey and Johnson [86] that parameters associated with different parts of the input can interact in a wide variety of ways in producing nonpolynomial complexity. Downey and Fellows =-=[47]-=- initiated a systematic analysis of the complexity of parameterized decision problems. Specifically, one of the principle ideas of parameterized complexity is to look more deeply into the structure of... |

802 | Introduction to Algorithms (Second Edition
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Citation Context ...ransitive closure of its label D after insertion of one ordered pair v < w (or w < v). Updating the transitive closure after one insertion can be done in O(|B| 2 ) time (see problem 25-1, page 641 in =-=[27]-=-). These updates are needed to generate the labels for the children, of which there are at most two. Thus the time taken in the third step of the 1-Sided Crossing Minimization algorithm is O(φ k · |B|... |

550 | Graph Drawing
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Citation Context ...ak of an X × Y × Z drawing with volume X · Y · Z. That is, the volume of a 3D drawing is the number of gridpoints in the bounding box. Minimising the volume in 3D drawings is a widely studied problem =-=[15, 17, 18, 19, 21, 22, 25, 26, 28, 29, 33, 39, 48, 53]-=-. The following general bounds are known for the volume of 3D drawings in terms of the track-number. Other papers to employ track layouts in the production of 3D drawings include [19, 22, 29, 33, 39].... |

445 | Combinatorial algorithms for integrated circuit layout - Lengauer - 1990 |

393 |
Some simplified NP-complete graph problems
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Citation Context ...s a component isomorphic to G, and thus G is k-colourable. The next result follows from the reduction in Theorem 3 and since it is NP-complete to determine if a 4-regular planar graph is 3-colourable =-=[19, 42]-=-. Corollary 1. It is NP-complete to determine if a given 4-regular planar graph G has arch-number an(G) ≤ 2. 5 Extremal Questions In this section we consider the extremal questions: • what is the maxi... |

351 |
Graph Drawing: Algorithms for the Visualization of Graphs
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- 1999
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Citation Context ...computing queue layouts [110, 165]. These ideas are made particularly concrete in the case of trees (see Lemmas 5.16 and 5.17). 1.1.2 3D graph drawings Graph drawing in the plane is well-studied (see =-=[35, 126]-=-). Motivated by experimental evidence suggesting that displaying a graph in three dimensions is better than in two [193, 194], and applications including information visualization [193], VLSI circuit ... |

313 |
Graph Classes: A Survey
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- 1999
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Citation Context ...o of which form an X-crossing. Thus the partition into k chains defines the desired edge colouring of a (k,t)-track layout. Note that Lemma 2 essentially says that permutation graphs are perfect (see =-=[16]-=-). 2.2 Extremal Questions Consider the maximum number of edges in a track layout. It follows from Lemma 1 that an n-vertex 2-track graph has at most n − 1 edges, which generalises to (k,2)-track graph... |

275 |
A decomposition theorem for partially ordered sets
- Dilworth
- 1950
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Citation Context ... E(G[A, B]). Two edges are unrelated under � if and only if they form an X-crossing. Thus an antichain in � is a crossing tuple. By assumption, � has no antichain of size k + 1. By Dilworth’s Theorem =-=[24]-=-, E(G[A, B]) can be partitioned into k chains. A chain in � is a set of edges of G[A, B], no two of which form an X-crossing. Thus the partition into k chains defines the desired edge colouring of a (... |

260 | A partial k-arboretum of graphs with bounded treewidth, Theoret
- Bodlaender
- 1998
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Citation Context ...APTER 6. LAYOUTS OF BOUNDED TREEWIDTH GRAPHS 77 are precisely the forests. Graphs with treewidth at most two are called series-parallel 1 , and are characterized as those graphs with no K4 minor (see =-=[14]-=-). A k-tree for some k ∈ N is defined recursively as 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 ... |

254 |
Graph rewriting: an algebraic and logic approach, Handbook of Theoretical Computer Science
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Citation Context ... powerful tool for deriving FPT algorithms. If a graph has bounded treewidth, then many intractable problems become tractable by dynamic programming on tree-decompositions, and by automata techniques =-=[30]-=-. These FPT methods, relying on tree-decompositions of graphs, are currently “approaching” practicality. The best algorithm for computing a tree-decomposition is due to Bodlaender [13]. The algorithm ... |

242 |
A combinatorial problem in geometry
- Erdös, Szekeres
(Show Context)
Citation Context ...ck. Let {e1, e2, . . . , eq} be a maximum rainbow in σ, where ei is nested inside ei+1 for all i < q. Let xi be the division vertex of G ′ corresponding to each edge ei. By the Erdös-Szekeres Theorem =-=[77]-=-, the permutation of {x1, . . . , xq} in the second track has an increasing or decreasing subsequence (with respect to the indices) of at least √ q vertices. Depending on whether it is increasing or d... |

217 | Crossing number is NP-complete - Garey, Johnson - 1983 |

207 |
Incidence matrices and interval graphs
- Fulkerson, Gross
- 1965
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Citation Context ...uivalent is due to Robertson and Seymour [169]. That (b) and (d) are equivalent is the characterization of chordal graphs in terms of ‘perfect elimination’ vertex-orderings due to Fulkerson and Gross =-=[81]-=-. 6.1.2 Tree-partitions As in the definition of a tree-decomposition, let G be graph and let {Tx ⊆ V (G) : x ∈ V (T )} be a partition of V (G) into subsets (called bags) indexed by the nodes of a tree... |

202 |
A linear-time algorithm for finding tree-decompositions of small treewidth
- Bodlaender
- 1996
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Citation Context ...mata techniques [30]. These FPT methods, relying on tree-decompositions of graphs, are currently “approaching” practicality. The best algorithm for computing a tree-decomposition is due to Bodlaender =-=[13]-=-. The algorithm runs in O(232k3|G|) time, where G is an input graph and k is an upper bound on its treewidth. There is a steadily growing list of examples of FPT algorithms with more practical costs i... |

182 |
Space Efficient Static Trees and Graphs
- Jacobson
- 1989
(Show Context)
Citation Context ...ude sorting permutations [34, 47, 83, 86, 99], fault tolerant VLSI design [17, 89–91], complexity theory [36, 37, 63], compact graph 1http://www.emba.uvm.edu/~archdeac/problems/stackq.htm 3 encodings =-=[61, 79]-=-, compact routing tables [43], and graph drawing [6, 24, 105, 106]. Numerous other aspects of stack layouts have been studied in the literature [7, 8, 10, 11, 14–16, 18, 20, 22, 32, 33, 35, 38, 40, 49... |

172 | Partially ordered sets - Dushnik, Miller - 1941 |

160 |
Proskurowski: Linear time algorithms for NP-hard problems restricted to partial k-trees
- Arnborg, A
- 1989
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Citation Context ...nded by its treewidth (Corollary 6.2). Treewidth, first defined by Halin [100], although largely unnoticed until independently rediscovered by Robertson and Seymour [169] and Arnborg and Proskurowski =-=[4]-=-, is a measure of the similarity of a graph to a tree (see Section 6.1.1 for the definition). Many graphs arising in applications of graph drawing do have small treewidth. Outerplanar and series-paral... |

159 |
How to draw a planar graph on a grid
- FRAYSSEIX, PACH, et al.
- 1990
(Show Context)
Citation Context ... been considered include symmetry [115–118], aspect ratio [24, 90], angular resolution [24, 90], edge-separation [24, 90], and convexity [23, 24, 64, 180]. The classical result of de Fraysseix et al. =-=[34]-=- and Schnyder [176] states that every planar graph has an O(n 2 ) area 2D straight-line grid drawing. In contrast to the case in the plane, a folklore result states that every graph has a 3D drawing. ... |

154 | Vertex cover: Further observations and further improvements
- Chen, Kanj, et al.
(Show Context)
Citation Context ... only on parameter k. A problem in FPT is thus solvable in polynomial time for a fixed k. The classical example is the FPT algorithm that solves the vertex cover problem in time O(kn + 1.29 k · k 2 ) =-=[22, 46]-=-. So the problem is well solved for input graphs of any size so long as k is no more than around 100. Yet it is not surprising that many parameterized problems appear not to be in FPT . For instance, ... |

144 | Succinct representation of balanced parentheses, static trees and planar graphs
- Munro, Raman
- 1997
(Show Context)
Citation Context ...ude sorting permutations [34, 47, 83, 86, 99], fault tolerant VLSI design [17, 89–91], complexity theory [36, 37, 63], compact graph 1http://www.emba.uvm.edu/~archdeac/problems/stackq.htm 3 encodings =-=[61, 79]-=-, compact routing tables [43], and graph drawing [6, 24, 105, 106]. Numerous other aspects of stack layouts have been studied in the literature [7, 8, 10, 11, 14–16, 18, 20, 22, 32, 33, 35, 38, 40, 49... |

131 |
On straight lines representation of planar graphs
- Fáry
- 1948
(Show Context)
Citation Context ... n-vertex planar graph G has a subdivision D with at most n − 2 division vertices per edge such that D admits an n-track layout with every edge having span one. Proof. By the classical result of Fáry =-=[79]-=- and Wagner [192], G has a straight-line plane drawing. Rotate such a drawing so that every vertex has a unique Y -coordinate. Draw n lines parallel to the X-axis, one through each vertex, and subdivi... |

121 |
The complexity of coloring circular arcs and chords
- Garey, Johnson, et al.
- 1980
(Show Context)
Citation Context ...ot true. There exists vertex orderings with no (k + 1)-edge twist that require Ω(k log k) stacks [67]. Moreover, it isNP-complete to test if a fixed vertex ordering of a graph admits a k-stack layout =-=[41]-=-2. On the other hand, Kostochka [68] proved that a vertex ordering with no 3-edge twist admits a 5-stack layout, and Ageev [1] proved that 5-stacks are sometimes necessary in this case. In general, Ko... |

111 |
On acyclic colorings of planar graphs
- Borodin
- 1979
(Show Context)
Citation Context ... chromatic number of a graph G, denoted by χa(G), is the minimum number of colours in an acyclic vertex colouring of G. This concept was introduced by Grünbaum [36], and has since been widely studied =-=[3, 4, 5, 10, 11, 12, 13, 14, 14, 16, 34, 35, 43, 47]-=-. By Lemma 1, each 2-track subgraph in an (edge-monochromatic) track layout is a forest of caterpillars. Thus the underlying vertex colouring is acyclic. Hence, χa(G) ≤ tn(G) . (1) A number of the res... |

90 |
The book thickness of a graph
- Bernhart, Kainen
- 1979
(Show Context)
Citation Context ...of a graph G, denoted by qn(G), is the minimum k such that G is a k-queue graph. See our companion paper [30] for a list of references and applications of stack and queue layouts. Bernhart and Kainen =-=[6]-=- observed that the 1-stack graphs are precisely the outerplanar graphs, and that 2-stack graphs are characterised as the subgraphs of planar Hamiltonian graphs. Heath and Rosenberg [41] characterised ... |

89 |
Acyclic colorings of planar graphs
- Grünbaum
- 1973
(Show Context)
Citation Context ...ives at least three colours. The acyclic chromatic number of a graph G, denoted by χa(G), is the minimum number of colours in an acyclic vertex colouring of G. This concept was introduced by Grünbaum =-=[36]-=-, and has since been widely studied [3, 4, 5, 10, 11, 12, 13, 14, 14, 16, 34, 35, 43, 47]. By Lemma 1, each 2-track subgraph in an (edge-monochromatic) track layout is a forest of caterpillars. Thus t... |

87 |
Edge Crossings in Drawings of Bipartite Graphs
- Eades, Wormald
- 1994
(Show Context)
Citation Context ... (that is, the problem becomes discrete), choosing vertex orderings that minimize the number of edge crossings in layered drawings is in fact an N P-complete problem even if there are only two layers =-=[70]-=-. The two layer problem was proposed by Harary [102], Harary and Schwenk [103] and Watkins [198]. They gave the first structural results for the problem. Two-layer drawings are of fundamental importan... |

86 | On the Computational Complexity of Upward and Rectilinear Planarity Testing - Garg, Tamassia - 1995 |

79 | Restricted colorings of graphs
- Alon
- 1993
(Show Context)
Citation Context ... chromatic number of a graph G, denoted by χa(G), is the minimum number of colours in an acyclic vertex colouring of G. This concept was introduced by Grünbaum [36], and has since been widely studied =-=[3, 4, 5, 10, 11, 12, 13, 14, 14, 16, 34, 35, 43, 47]-=-. By Lemma 1, each 2-track subgraph in an (edge-monochromatic) track layout is a forest of caterpillars. Thus the underlying vertex colouring is acyclic. Hence, χa(G) ≤ tn(G) . (1) A number of the res... |

69 | Parameterized complexity: A framework for systematically confronting computational intractability
- Downey, Fellows, et al.
- 1997
(Show Context)
Citation Context ... only on parameter k. A problem in FPT is thus solvable in polynomial time for a fixed k. The classical example is the FPT algorithm that solves the vertex cover problem in time O(kn + 1.29 k · k 2 ) =-=[22, 46]-=-. So the problem is well solved for input graphs of any size so long as k is no more than around 100. Yet it is not surprising that many parameterized problems appear not to be in FPT . For instance, ... |

69 | 2-layer straightline crossing minimization: Performance of exa ct and heuristic algorithms - Juenger, Mutzel - 1997 |

66 | How to draw a planar graph on a grid. Combinatorica 10:41–51 - Fraysseix, Pach, et al. - 1990 |

53 | The complexity of wire–routing and finding minimum area layouts for arbitrary VLSI circuits - Kramer, Leeuwen - 1984 |

52 | Embedding graphs in books: a layout problem with applications to VLSI design
- CHUNG, LEIGHTON, et al.
- 1987
(Show Context)
Citation Context ...me algorithm for the analogous problem for queue layouts.sCHAPTER 1. INTRODUCTION 4 A tree is a 1-queue graph, since in a breadth-first vertex ordering of a tree no two edges are nested. Chung et al. =-=[25]-=- proved that in a depth-first vertex ordering of a tree no two edges cross. Thus trees are 1-stack graphs. Loosely speaking, treewidth measures how similar a graph is to a tree, and band-width is a me... |

49 |
Decomposition of finite graphs into forests
- Nash-Williams
- 1964
(Show Context)
Citation Context ... bounded by stack-number. In particular, every k-stack graph G has acyclic chromatic number χa(G) ≤ 80 k(2k−1) . Proof. The edges of an outerplanar graph can be partitioned into two acyclic subgraphs =-=[45]-=-. Thus G has a 2k-stack layout in which each stack is acyclic. The result follows from Theorem 4. Note that the converse of Theorem 5 is not true. Blankenship and Oporowski [7, 8, 9] proved that the s... |

48 |
Laying out graphs using queues
- HEATH, ROSENBERG
- 1992
(Show Context)
Citation Context ...hart and Kainen [6] observed that the 1-stack graphs are precisely the outerplanar graphs, and that 2-stack graphs are characterised as the subgraphs of planar Hamiltonian graphs. Heath and Rosenberg =-=[41]-=- characterised 1-queue graphs as the ‘arched levelled planar’ graphs. The following lemma highlights the fundamental relationship between track layouts, and queue and stack layouts. Its proof follows ... |

48 |
Automatic display of hierarchized graphs for computer aided decision analysis
- Carpano
- 1980
(Show Context)
Citation Context ...NTRODUCTION 7 1.1.3 Layered (hierarchical) drawings A common method for drawing directed graphs, which produces layered drawings or hierarchical drawings was introduced by Tomii et al. [188], Carpano =-=[20]-=- and Sugiyama et al. [183]. In this type of drawing, vertices are arranged on h ≥ 2 layers (that is, on h parallel lines in the plane), and edges are drawn as straight line-segments between vertices o... |

47 | Acyclic and oriented chromatic numbers of graphs
- Kostochka, Sopena, et al.
- 1997
(Show Context)
Citation Context ...er bound the acyclic chromatic number by various ‘geometric’ graph parameters. Many other variations of the chromatic number (including star chromatic number [1, 35, 36] and oriented chromatic number =-=[44, 50, 52]-=-) are bounded by the acyclic chromatic number. Thus our results also apply to these other types of colourings—we omit the details. Alon and Marshall [2] proved the following application of acyclic col... |

44 | Three-dimensional circuit layouts - Leighton, Rosenberg - 1986 |

43 | Straight-line drawings on restricted integer grids in two and three dimensions
- FELSNER, LIOTTA, et al.
(Show Context)
Citation Context ...eries-parallel graphs 15 Di Giacomo et al. [20] Halin 8 † Di Giacomo and Meijer [22] X-trees 6 † Di Giacomo and Meijer [22] outerplanar 5 † Lemma 22 1-queue graphs 4 Theorem 11 trees 3 Felsner et al. =-=[33]-=- Y − 1 and Z − 1, then we speak of an X × Y × Z drawing with volume X · Y · Z. That is, the volume of a 3D drawing is the number of gridpoints in the bounding box. Minimising the volume in 3D drawings... |

41 |
Computing permutations with double-ended queues, parallel stacks and parallel queues
- PRATT
- 1973
(Show Context)
Citation Context ...ck layouts are more commonly called book embeddings, and stack-number has been called bookthickness, fixed outer-thickness, and page-number. Applications of stack layouts include sorting permutations =-=[34, 47, 83, 86, 99]-=-, fault tolerant VLSI design [17, 89–91], complexity theory [36, 37, 63], compact graph 1http://www.emba.uvm.edu/~archdeac/problems/stackq.htm 3 encodings [61, 79], compact routing tables [43], and gr... |

36 |
Comparing queues and stacks as mechanisms for laying out graphs
- HEATH, LEIGHTON, et al.
- 1992
(Show Context)
Citation Context ...of a graph G, denoted by sn(G) (qn(G), an(G)), is the minimum k such that G is a k-stack (k-queue, k-arch) graph. Stack and queue layouts were respectively introduced by Ollmann [82] and Heath et al. =-=[53, 57]-=-. As far as we are aware, arch layouts have not previously been studied, although Dan Archdeacon1 suggests doing so. Stack layouts are more commonly called book embeddings, and stack-number has been c... |

36 | Twarog, “3D Graph Drawing with Simulated Annealing - Cruz, P - 1995 |

34 |
On an edge crossing problem
- Eades, McKay, et al.
- 1986
(Show Context)
Citation Context ...anar graphs are easily characterized, and there is a simple linear-time algorithm to recognize biplanar graphs. The next lemma recalls and augments the characterization given in Lemma 2.4. Lemma 4.1 (=-=[66, 103, 188]-=-). Let G be a graph. The following are equivalent: (a) G is biplanar. (b) G is a forest of caterpillars (see Figure 4.2). (c) G is acyclic and contains no 2-claw. (d) The graph obtained from G by dele... |

33 | Star coloring of graphs
- Fertin, Raspaud, et al.
- 2004
(Show Context)
Citation Context ... chromatic number of a graph G, denoted by χa(G), is the minimum number of colours in an acyclic vertex colouring of G. This concept was introduced by Grünbaum [36], and has since been widely studied =-=[3, 4, 5, 10, 11, 12, 13, 14, 14, 16, 34, 35, 43, 47]-=-. By Lemma 1, each 2-track subgraph in an (edge-monochromatic) track layout is a forest of caterpillars. Thus the underlying vertex colouring is acyclic. Hence, χa(G) ≤ tn(G) . (1) A number of the res... |

33 |
Problems from the world surrounding perfect graphs. Zastos
- GYÁRFÁS
- 1987
(Show Context)
Citation Context ... is bounded by β and β is bounded by α then α and β are tied. Clearly, if α and β are tied then a graph family G has bounded α if and only if G has bounded β. These notions were introduced by Gyárfás =-=[37]-=- in relation to near-perfect graph families for which the chromatic number is bounded by the clique-number. A vertex ordering of an n-vertex graph G is a bijection σ : V (G) → {1, 2, . . ., n}. We wri... |

33 |
Drawing Graphs in Two Layers
- Eades, Whitesides
- 1994
(Show Context)
Citation Context ...he problems, 1- and 2-LAYER PLANARIZATION have received less attention in the graph drawing literature than their crossing minimization counterparts. The 2-LAYER PLANARIZATION problem is N P-complete =-=[69, 188]-=-, even for planar biconnected bipartite graphs with vertices in respective bipartitions having degree two and three [69]. Eades and Whitesides [69] show that the 1-LAYER PLANARIZATION problem is also ... |

33 | On higher-dimensional orthogonal graph drawing - Wood - 1997 |