## On the Queue Number of Planar Graphs (2010)

Citations: | 3 - 0 self |

### BibTeX

@MISC{Frati10onthe,

author = {Fabrizio Frati and János Pach},

title = {On the Queue Number of Planar Graphs},

year = {2010}

}

### OpenURL

### Abstract

We prove that planar graphs have O(log 4 n) queue number, thus improving upon the previous O ( √ n) upper bound. Consequently, planar graphs admit 3D straight-line crossingfree grid drawings in O(n log c n) volume, for some constant c, thus improving upon the previous O(n 3/2) upper bound. 2 1

### Citations

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Citation Context ..., hence level-2connected graphs have O(log 2 n) queue number, thus planar graphs have O(log 4 n) queue number. Our result sheds new light on one of the most studied Graph Drawing problems (see, e.g., =-=[5, 14, 3, 10, 8, 6]-=-): Given an n-vertex planar graph which is the volume required to draw it in 3D, representing edges with straight-line segments that cross only at common endpoints? The previously best known upper bou... |

43 |
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Citation Context ...) and (w, z) nest if u < w < z < v or w < u < v < z), and the queue number is the minimum k such that G has a queue layout on k queues. Queue layouts were introduced by Heath, Leighton, and Rosenberg =-=[20, 24]-=-, motivated by applications, e.g., in parallel process scheduling [1], matrix-computations [28], and sorting permutations and networks [29, 32]. Computing the queue number of a graph is N P-complete. ... |

38 | Straight-line drawings on restricted integer grids in two and three dimensions
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Citation Context ..., hence level-2connected graphs have O(log 2 n) queue number, thus planar graphs have O(log 4 n) queue number. Our result sheds new light on one of the most studied Graph Drawing problems (see, e.g., =-=[5, 14, 3, 10, 8, 6]-=-): Given an n-vertex planar graph which is the volume required to draw it in 3D, representing edges with straight-line segments that cross only at common endpoints? The previously best known upper bou... |

37 |
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Citation Context ...ayouts were introduced by Heath, Leighton, and Rosenberg [20, 24], motivated by applications, e.g., in parallel process scheduling [1], matrix-computations [28], and sorting permutations and networks =-=[29, 32]-=-. Computing the queue number of a graph is N P-complete. Namely, it is known that deciding if a graph has queue number 1 is N P-complete [24]. However, from a combinatorial point of view, a large numb... |

33 |
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Citation Context ...) and (w, z) nest if u < w < z < v or w < u < v < z), and the queue number is the minimum k such that G has a queue layout on k queues. Queue layouts were introduced by Heath, Leighton, and Rosenberg =-=[20, 24]-=-, motivated by applications, e.g., in parallel process scheduling [1], matrix-computations [28], and sorting permutations and networks [29, 32]. Computing the queue number of a graph is N P-complete. ... |

30 | On linear layouts of graphs
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Citation Context ...nce also planar 3trees, have constant queue number. Observe that the problem of determining the queue number of planar graphs is cited into several papers and collections of open problems (see, e.g., =-=[20, 24, 6, 3, 7, 11, 8]-=-). In this paper, we prove that the queue number of planar graphs is O(log 4 n). The proof is constructive and is based on a polynomial-time algorithm that computes a queue layout with such a queue nu... |

30 |
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Citation Context ...e literature is rich of combinatorial and algorithmic contributions on the page number of various classes of graphs (see, e.g., [4, 18, 15, 19, 25, 26, 13, 12, 17, 16]). A famous result of Yannakakis =-=[33]-=- states that a planar graph has page number at most four. Queue layout and queue number are the dual concepts of book embedding and page number, respectively. A queue layout on k queues of a graph G(V... |

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Citation Context ...t have queue number at most e √ m [9], at most 3 w · 6 (4w −3w−1)/9 − 1 [8], at most 36∆w [8], at most p [8], at most ⌈b/2⌉ [24], and at most t − 1 [8], respectively. Queue layouts of directed graphs =-=[23, 22]-=- and posets [21] have also been studied. As in many graph problems, a special attention has been devoted to planar graphs and their subclasses. For example, trees have queue number 1 [24], outerplanar... |

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Citation Context ...width w, graphs with tree-width w and degree ∆, graphs with path-width p, graphs with bandwidth b, and graphs with track number t have queue number at most e √ m [9], at most 3 w · 6 (4w −3w−1)/9 − 1 =-=[8]-=-, at most 36∆w [8], at most p [8], at most ⌈b/2⌉ [24], and at most t − 1 [8], respectively. Queue layouts of directed graphs [23, 22] and posets [21] have also been studied. As in many graph problems,... |

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23 |
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Citation Context ...ayouts were introduced by Heath, Leighton, and Rosenberg [20, 24], motivated by applications, e.g., in parallel process scheduling [1], matrix-computations [28], and sorting permutations and networks =-=[29, 32]-=-. Computing the queue number of a graph is N P-complete. Namely, it is known that deciding if a graph has queue number 1 is N P-complete [24]. However, from a combinatorial point of view, a large numb... |

19 | Stack and queue layouts of posets
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Citation Context ...t most p [9], graphs with band-width b have queue-number at most ⌈b/2⌉ [25], and graphs with track number t have queue-number at most t − 1 [9]. Queue layouts of directed graphs [23], [24] and posets =-=[22]-=- have also been studied. As in many graph problems, a special attention has been devoted to planar graphs and their subclasses. For example, trees have queue number 1 [25], outerplanar graphs have que... |

18 | Three-dimensional grid drawings with subquadratic volume
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Citation Context ..., hence level-2connected graphs have O(log 2 n) queue number, thus planar graphs have O(log 4 n) queue number. Our result sheds new light on one of the most studied Graph Drawing problems (see, e.g., =-=[5, 14, 3, 10, 8, 6]-=-): Given an n-vertex planar graph which is the volume required to draw it in 3D, representing edges with straight-line segments that cross only at common endpoints? The previously best known upper bou... |

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Citation Context ...nce also planar 3trees, have constant queue number. Observe that the problem of determining the queue number of planar graphs is cited into several papers and collections of open problems (see, e.g., =-=[20, 24, 6, 3, 7, 11, 8]-=-). In this paper, we prove that the queue number of planar graphs is O(log 4 n). The proof is constructive and is based on a polynomial-time algorithm that computes a queue layout with such a queue nu... |

16 |
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Citation Context ... number is the minimum k such that G has a book embedding on k pages. The literature is rich of combinatorial and algorithmic contributions on the page number of various classes of graphs (see, e.g., =-=[4, 18, 15, 19, 25, 26, 13, 12, 17, 16]-=-). A famous result of Yannakakis [33] states that a planar graph has page number at most four. Queue layout and queue number are the dual concepts of book embedding and page number, respectively. A qu... |

15 |
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Citation Context ... number is the minimum k such that G has a book embedding on k pages. The literature is rich of combinatorial and algorithmic contributions on the page number of various classes of graphs (see, e.g., =-=[4, 18, 15, 19, 25, 26, 13, 12, 17, 16]-=-). A famous result of Yannakakis [33] states that a planar graph has page number at most four. Queue layout and queue number are the dual concepts of book embedding and page number, respectively. A qu... |

14 |
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Citation Context ...at G has a queue layout on k queues. Queue layouts were introduced by Heath, Leighton, and Rosenberg [20, 24], motivated by applications, e.g., in parallel process scheduling [1], matrix-computations =-=[28]-=-, and sorting permutations and networks [29, 32]. Computing the queue number of a graph is N P-complete. Namely, it is known that deciding if a graph has queue number 1 is N P-complete [24]. However, ... |

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Citation Context ... is the minimum k such that G has a queue layout on k queues. Queue layouts were introduced by Heath, Leighton, and Rosenberg [20, 24], motivated by applications, e.g., in parallel process scheduling =-=[1]-=-, matrix-computations [28], and sorting permutations and networks [29, 32]. Computing the queue number of a graph is N P-complete. Namely, it is known that deciding if a graph has queue number 1 is N ... |

11 |
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Citation Context ... number is the minimum k such that G has a book embedding on k pages. The literature is rich of combinatorial and algorithmic contributions on the page number of various classes of graphs (see, e.g., =-=[4, 18, 15, 19, 25, 26, 13, 12, 17, 16]-=-). A famous result of Yannakakis [33] states that a planar graph has page number at most four. Queue layout and queue number are the dual concepts of book embedding and page number, respectively. A qu... |

11 |
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Citation Context ... number is the minimum k such that G has a book embedding on k pages. The literature is rich of combinatorial and algorithmic contributions on the page number of various classes of graphs (see, e.g., =-=[4]-=-, [13], [14], [16], [17], [18], [19], [20], [26], [27]). A famous result of Yannakakis [34] states that a planar graph has page number at most four. Queue layout and queue number are the dual concepts... |

9 |
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Citation Context ...nce also planar 3trees, have constant queue number. Observe that the problem of determining the queue number of planar graphs is cited into several papers and collections of open problems (see, e.g., =-=[20, 24, 6, 3, 7, 11, 8]-=-). In this paper, we prove that the queue number of planar graphs is O(log 4 n). The proof is constructive and is based on a polynomial-time algorithm that computes a queue layout with such a queue nu... |

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Citation Context ...ention has been devoted to planar graphs and their subclasses. For example, trees have queue number 1 [24], outerplanar graphs have queue number 2 [20], and series-parallel graphs have queue number 3 =-=[30]-=-. However, for general planar graphs the best known upper bound for the queue number is O( √ n) (a consequence of the results on graphs with O(n) edges [24, 31, 9]), while no super-constant lower boun... |

8 |
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Citation Context ...artition of its edges such that all the elements of the partition enforce some specific property. Linear layouts play an important role in Graph Theory and their study goes back to 1973, when Ollmann =-=[27]-=- introduced the concept of book embedding (later also called stack layout) and book thickness (later also called stack number, fixed outer-thickness, and, most successfully, page number) of a graph. A... |

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Citation Context ...ber of a graph G is the minimum number of colors such that G admits a proper coloring in which each pair of colors induces a forest. Since every planar graph has acyclic chromatic number at most five =-=[2]-=- and queue number at most O(log 4 n) (by Theorem 2), then every planar graph has track number at most tn(G) ≤ O(log 4 n) 4 = O(log 16 n). Since every graph G has an O(tn(G)) × O(tn(G)) × O(n) volume 3... |

6 | On crossing sets, disjoint sets, and pagenumber
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Citation Context ... series-parallel graphs have queue number 3 [30]. However, for general planar graphs the best known upper bound for the queue number is O( √ n) (a consequence of the results on graphs with O(n) edges =-=[24, 31, 9]-=-), while no super-constant lower bound is known. Heath et al. [20, 24] conjectured that planar graphs have O(1) queue number. Pemmaraju [28] conjectured that a certain class of planar graphs, namely p... |

5 |
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Citation Context ...ber of a graph G is the minimum number of colors such that G admits a proper coloring in which each pair of colors induces a forest. Since every planar graph has acyclic chromatic number at most five =-=[2]-=- and queue number O(log 4 n) (by Theorem 2), then every planar graph has track number O(log 16 n). Dujmović and Wood [11] proved that every c-colorable graph G with n vertices and track-number tn(G) ≤... |