## Planar graphs, via well-orderly maps and trees (2004)

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Venue: | In 30 th International Workshop, Graph - Theoretic Concepts in Computer Science (WG), volume 3353 of Lecture Notes in Computer Science |

Citations: | 19 - 4 self |

### BibTeX

@INPROCEEDINGS{Bonichon04planargraphs,,

author = {Nicolas Bonichon and Cyril Gavoille and Nicolas Hanusse and Dominique Poulalhon and Gilles Schaeffer},

title = {Planar graphs, via well-orderly maps and trees},

booktitle = {In 30 th International Workshop, Graph - Theoretic Concepts in Computer Science (WG), volume 3353 of Lecture Notes in Computer Science},

year = {2004},

pages = {270--284},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

Abstract. The family of well-orderly maps is a family of planar maps with the property that every connected planar graph has at least one plane embedding which is a well-orderly map. We show that the number of well-orderly maps with n nodes is at most 2 αn+O(log n) , where α ≈ 4.91. A direct consequence of this is a new upper bound on the number p(n) of unlabeled planar graphs with n nodes, log 2 p(n) � 4.91n. The result is then used to show that asymptotically almost all (labeled or unlabeled), (connected or not) planar graphs with n nodes have between 1.85n and 2.44n edges. Finally we obtain as an outcome of our combinatorial analysis an explicit linear time encoding algorithm for unlabeled planar graphs using, in the worst-case, a rate of 4.91 bits per node and of 2.82 bits per edge. 1

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Citation Context ...ameters k or k ′ . Finally root nodes that contribute to the parameters are pictured in a box. 3.3 Generating Functions of Trees and the Asymptotic Number of Well-Orderly Maps The reader can refer to =-=[25]-=- for a general presentation of enumeration of decomposable structures using grammars and generating series.s10 Nicolas Bonichon et al. = + = + = + = + + + + + Fig. 6. A decomposition of colored trees ... |

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Citation Context ... a lot of attention is given to efficiently represent discrete objects. At least two field of applications of high interests are concerned with succinct planar graph representation: Computer Graphics =-=[8,9,10]-=- and Networking [11,12,13,14]. 1.1 Related Works Obviously, without sharp asymptotic formula, properties and behavior of large random objects cannot be described precisely. For lack of an adequate mod... |

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Citation Context ... on the 4-page embedding of planar graphs (see [20]). In a series of articles, Lu et al. [21,22] refined the coding to 4m/3 + 5n thanks to orderly spanning trees, a generalization of Schnyder’s trees =-=[23]-=-. 1.2 Our Results Any planar embedding of a n-node planar graph with n nodes can be seen as a subgraph of a n-node triangulation of the plane. Once given a triangulation and a set of edges to keep (or... |

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Citation Context ... Succinct representation of n-node m-edge planar graphs has a long history. Turán [17] pioneered a 4m bit encoding, that has been improved later by Keeler and Westbrook [18] to 3.58m. Munro and Raman =-=[19]-=- then proposed a 2m + 8n bit encoding based on the 4-page embedding of planar graphs (see [20]). In a series of articles, Lu et al. [21,22] refined the coding to 4m/3 + 5n thanks to orderly spanning t... |

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Citation Context ... a lot of attention is given to efficiently represent discrete objects. At least two field of applications of high interests are concerned with succinct planar graph representation: Computer Graphics =-=[8,9,10]-=- and Networking [11,12,13,14]. 1.1 Related Works Obviously, without sharp asymptotic formula, properties and behavior of large random objects cannot be described precisely. For lack of an adequate mod... |

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Citation Context ... a lot of attention is given to efficiently represent discrete objects. At least two field of applications of high interests are concerned with succinct planar graph representation: Computer Graphics =-=[8,9,10]-=- and Networking [11,12,13,14]. 1.1 Related Works Obviously, without sharp asymptotic formula, properties and behavior of large random objects cannot be described precisely. For lack of an adequate mod... |

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Citation Context ...ed later by Keeler and Westbrook [18] to 3.58m. Munro and Raman [19] then proposed a 2m + 8n bit encoding based on the 4-page embedding of planar graphs (see [20]). In a series of articles, Lu et al. =-=[21,22]-=- refined the coding to 4m/3 + 5n thanks to orderly spanning trees, a generalization of Schnyder’s trees [23]. 1.2 Our Results Any planar embedding of a n-node planar graph with n nodes can be seen as ... |

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Citation Context ...iven to efficiently represent discrete objects. At least two field of applications of high interests are concerned with succinct planar graph representation: Computer Graphics [8,9,10] and Networking =-=[11,12,13,14]-=-. 1.1 Related Works Obviously, without sharp asymptotic formula, properties and behavior of large random objects cannot be described precisely. For lack of an adequate model, very little is known on r... |

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Citation Context ...d case 1.70n and 2.54n, by [5]. Succinct representation of n-node m-edge planar graphs has a long history. Turán [17] pioneered a 4m bit encoding, that has been improved later by Keeler and Westbrook =-=[18]-=- to 3.58m. Munro and Raman [19] then proposed a 2m + 8n bit encoding based on the 4-page embedding of planar graphs (see [20]). In a series of articles, Lu et al. [21,22] refined the coding to 4m/3 + ... |

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Citation Context ...-Orderly Maps and Trees 7 |Rn,ℓ| n−3 � ℓ=max{1,2n−m−6} 3 Counting and Coding Trees � � n + ℓ � |Hn,m|. m − 2n + 6 + ℓ |Rn,ℓ| � � n + ℓ . m − 2n + 6 + ℓ In this section we briefly recall a result from =-=[24]-=- about minimal realizers and plane trees. An encoding of well-orderly maps follows. 3.1 Minimal Realizers and Plane Trees A tree is planted if it is rooted on a leaf. Let Bn be the set of planted plan... |

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Citation Context ...ed later by Keeler and Westbrook [18] to 3.58m. Munro and Raman [19] then proposed a 2m + 8n bit encoding based on the 4-page embedding of planar graphs (see [20]). In a series of articles, Lu et al. =-=[21,22]-=- refined the coding to 4m/3 + 5n thanks to orderly spanning trees, a generalization of Schnyder’s trees [23]. 1.2 Our Results Any planar embedding of a n-node planar graph with n nodes can be seen as ... |

31 |
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Citation Context ...ed a 4m bit encoding, that has been improved later by Keeler and Westbrook [18] to 3.58m. Munro and Raman [19] then proposed a 2m + 8n bit encoding based on the 4-page embedding of planar graphs (see =-=[20]-=-). In a series of articles, Lu et al. [21,22] refined the coding to 4m/3 + 5n thanks to orderly spanning trees, a generalization of Schnyder’s trees [23]. 1.2 Our Results Any planar embedding of a n-n... |

30 | N.: Compact routing tables for graphs of bounded genus
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Citation Context ...iven to efficiently represent discrete objects. At least two field of applications of high interests are concerned with succinct planar graph representation: Computer Graphics [8,9,10] and Networking =-=[11,12,13,14]-=-. 1.1 Related Works Obviously, without sharp asymptotic formula, properties and behavior of large random objects cannot be described precisely. For lack of an adequate model, very little is known on r... |

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Citation Context ...lanar graphs has been investigated in the last decade. Using a simple Markov chain, Denis et al. [2] showed, that, experimentally, random labeled planar graphs have 2n edges. In fact, Bodirsky et al. =-=[15]-=- have designed the first polynomial time (uniform) random generator of labeled planar graphs. Although limited in their experiments (mainly by the time complexity of this algorithm), they showed that ... |

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Citation Context ...asymptotic on the number of labeled planar graphs. This asymptotic is on the form n!λ n+o(n) [2,3], and in [4], a precise estimation of λ is given: 27.2268 < λ < 27.2269. The upper bound on µ, due to =-=[5]-=-, comes from succinct encoding of plane planar graphs. More precisely, after a suitable embedding and triangulation of the planar graph, it is shown that such embeddings can be represented by a binary... |

18 |
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Citation Context ...is algorithm), they showed that actually the number of edges in a random labeled planar graph is more than 2n. The best proved bounds on the number of edges in a random labeled planar graph are 1.85n =-=[16]-=- and 2.54n [5], for the unlabeled case 1.70n and 2.54n, by [5]. Succinct representation of n-node m-edge planar graphs has a long history. Turán [17] pioneered a 4m bit encoding, that has been improve... |

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Citation Context ... of numbers p(n) of unlabeled planar graphs. This growth rate, defined as µ = limn→∞ p(n) 1/n , currently ranges between 27.2268 and 32.1556 (a superadditivity argument shows that such a limit exists =-=[2,3]-=-). The lower bound on µ comes from an asymptotic on the number of labeled planar graphs. This asymptotic is on the form n!λ n+o(n) [2,3], and in [4], a precise estimation of λ is given: 27.2268 < λ < ... |

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Citation Context |

10 |
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Citation Context ...icult than counting the labeled version. And, as pointed out in [6], almost all labeled 2- and 1-connected planar graphs have exponentially large automorphism groups. In other words, Wright’s Theorem =-=[7]-=- does not hold for random planar graphs, the asymptotic number of labeled and unlabeled planar graphs differ in more than the n! term, i.e., λ < µ. So, an asymptotic on the number of labeled planar gr... |

8 |
Wagner’s theorem on realizers
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Citation Context ... clearly true for the two other trees of the realizer. Since for any realizer, ℓ0 + ℓ1 + ℓ2 + ∆ = 2n − 5, where ℓi is the number of leaves in Ti and ∆ is the number of 3-colored faces of the realizer =-=[26]-=-, the second result comes directely. ⊓⊔ In [27] a straight-line drawing algorithm base on minimal realizers is presented. This algorithm first computes the minimal realizer of a triangulation of the g... |

6 |
Optimal area algorithm for planar polyline drawings
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Citation Context ...verage complexity of such drawings: Corollary 1. The average grid size required (i.e., the average width and the average height) to draw a triangulation is at most ( 7n 7n 8 + o(n)) × ( 8 + o(n)). In =-=[28]-=- a polyline drawing algorithm also based on minimal realizers is proposed. The graph is then drawn on a grid (n− � � ℓ 2 −1)×ℓ, where ℓ is the number of leaves of the tree T0 of the obtained minimal r... |

5 |
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Citation Context ...dditivity argument shows that such a limit exists [2,3]). The lower bound on µ comes from an asymptotic on the number of labeled planar graphs. This asymptotic is on the form n!λ n+o(n) [2,3], and in =-=[4]-=-, a precise estimation of λ is given: 27.2268 < λ < 27.2269. The upper bound on µ, due to [5], comes from succinct encoding of plane planar graphs. More precisely, after a suitable embedding and trian... |

5 |
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Citation Context ...alizer. Since for any realizer, ℓ0 + ℓ1 + ℓ2 + ∆ = 2n − 5, where ℓi is the number of leaves in Ti and ∆ is the number of 3-colored faces of the realizer [26], the second result comes directely. ⊓⊔ In =-=[27]-=- a straight-line drawing algorithm base on minimal realizers is presented. This algorithm first computes the minimal realizer of a triangulation of the graph. Then the graph is drawn on a grid (n − 1 ... |

4 |
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Citation Context ...of 4.91 bits per node and of 2.82 bits per edge. 1 Introduction Counting the number of (non-isomorphic) planar graphs with n nodes is a wellknown long-standing unsolved graph-enumeration problem (cf. =-=[1]-=-). There is no known closed formula, neither asymptotic for unlabeled planar graphs. There are only upper and lower bounds on the growth rate of the sequence of numbers p(n) of unlabeled planar graphs... |

3 | Aspects algorithmiques et combinatoires des réaliseurs des graphes plans maximaux - Bonichon - 2002 |

2 |
D.J.: Random planar graphs (2001
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(Show Context)
Citation Context ... of numbers p(n) of unlabeled planar graphs. This growth rate, defined as µ = limn→∞ p(n) 1/n , currently ranges between 27.2268 and 32.1556 (a superadditivity argument shows that such a limit exists =-=[2,3]-=-). The lower bound on µ comes from an asymptotic on the number of labeled planar graphs. This asymptotic is on the form n!λ n+o(n) [2,3], and in [4], a precise estimation of λ is given: 27.2268 < λ < ... |