## Canonical Decomposition of Outerplanar Maps and Application to Enumeration, Coding and Generation (2003)

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Citations: | 12 - 1 self |

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@MISC{Bonichon03canonicaldecomposition,

author = {Nicolas Bonichon and Cyril Gavoille and Nicolas Hanusse},

title = {Canonical Decomposition of Outerplanar Maps and Application to Enumeration, Coding and Generation},

year = {2003}

}

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### Abstract

In this article we define a canonical decomposition of rooted outerplanar maps into a spanning tree and a list of edges. This decomposition, constructible in linear time, implies the existence of bijection between rooted outerplanar maps with n nodes and bicolored rooted ordered trees with n nodes where all the nodes of the last branch are colored white. As a consequence, for rooted outerplanar maps of n nodes, we derive: an enumeration formula, and an asymptotic of 2 3n (log n) ; an optimal data structure of asymptotically 3n bits, built in O(n) time, supporting adjacency and degree queries in worst-case constant time and neighbors query of a d-degree node in worst-case O(d) time...

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Citation Context ...), the cardinality of B (v i ) (see Lemma 5) and of B> (v i ) has a front edge. Starting from a connected outerplanar graph, we can compute a rooted outerplanar map using the algorithm presented in [=-=CNAO85]-=-. So the previous results on outerplanar maps can also be applied to outerplanar graphs. 5 Uniform Random Generation To randomly generate an outerplanar map, one can randomly generate a bicolored tree... |

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Citation Context ...des, or with a given number of nodes and of edges. 2 The Well-Orderly Tree of an Outerplanar Map In [BGH03] the authors introduced the well-orderly trees, a special case of the orderly spanning trees =-=[CLL01]-=-. Let T be a rooted spanning tree of a planar map H. Two nodes are unrelated if neither of them is an ancestor of the other in T . An edge of H is unrelated if its endpoints are unrelated. Let v 1 ; v... |

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Citation Context ...page consisting of non-intersecting edges. An outerplanar graph with a 1-page embedding is depicted on Fig. 1c). A planar map is a connected graph drawn on the sphere with non-intersecting edges (see =-=[CM92]-=- for a survey). A planar map is outerplanar if all the nodes lie on one face, called the outerface. For convenience, outerplanar maps are drawn on the plane such that the outerface corresponds to the ... |

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Citation Context ...plane with non-intersecting edges such that all the nodes lie on the boundary of the innite face, also called outerface. Characterization of outerplanar graphs has been given by Chartrand and Harary [=-=CH67]-=-: a graph is outerplanar if and only if it has neither K 2;3 nor K 4 as a minor. A linear time recognition algorithm has been given by Mitchell [Mit79]. Labeled and unlabeled outerplanar graphs can be... |

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Citation Context ...nar graphs has been given by Chartrand and Harary [CH67]: a graph is outerplanar if and only if it has neither K 2;3 nor K 4 as a minor. A linear time recognition algorithm has been given by Mitchell =-=[Mit79]-=-. Labeled and unlabeled outerplanar graphs can be randomly generated in O(n 4 log n) space and O(n 2 ) time [BK03] after a preprocessing of O(n 5 ) time. Among graph properties, outerplanar graphs con... |

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Citation Context ..., a random 1 In their article, 1-page embeddings are called non-crossing graphs. 2 outerplanar map can be generated uniformly with O(n) space and O(n 2 ) average time. Using Floating-Point Arithmetic =-=[DZ99-=-], this average time complexity can be reduced to O(n 1+ ). In Section 5, we propose a O(n) expected time and O(n) space complexity generating algorithm. It can generate outerplanar maps with a given ... |

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Citation Context ...n) space complexity generating algorithm. It can generate outerplanar maps with a given number of nodes, or with a given number of nodes and of edges. 2 The Well-Orderly Tree of an Outerplanar Map In =-=[BGH03]-=- the authors introduced the well-orderly trees, a special case of the orderly spanning trees [CLL01]. Let T be a rooted spanning tree of a planar map H. Two nodes are unrelated if neither of them is a... |

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Citation Context ...orm random generation on the set ~ M n (or on ~ M n;m ), and thus on the corresponding outerplanar maps. A rooted tree can be generated in linear time using for example the Arnold and Sleep algorithm =-=[AS80-=-]. The colors of the n 2 nodes are colored black with probability 1=2 (recall that the root and the last leaf are forced to be colored white by denition of B n ). This provides an O(n) time algorithm ... |

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Citation Context ...pace and O(n 2 ) time [BK03] after a preprocessing of O(n 5 ) time. Among graph properties, outerplanar graphs contain trees, have tree-width at most two, and are exactly the graphs of pagenumber one =-=[Bil92]-=-. Recall that a graph G has pagenumber k if k is the smaller integer for which G has a k-page embedding, also called book embedding. In such an embedding the nodes are drawn on a straight line (the sp... |

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Citation Context ...n answered in worst-case constant time. Using a grammar to produce bicolored rooted ordered trees with n nodes where all the nodes of the last branch are colored white, and using Goldwurm's algorithm =-=[Gol95]-=-, a random 1 In their article, 1-page embeddings are called non-crossing graphs. 2 outerplanar map can be generated uniformly with O(n) space and O(n 2 ) average time. Using Floating-Point Arithmetic ... |

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Citation Context ...d=1 w d = 16 9 : We have: T n;1 = 1 2n 3 2n 3 n 2 = 1 n 2(n 2) n 2 2 2n 16n p n : The result follows. We now rely Narayana numbers with biconnected outerplanar maps. The Narayana numbers [Nar59] count rooted plane trees with n nodes and ` leaves: Y n;` = 1 n n ` n 2 n ` 2 ; for all n > ` > 0: Biconnected outerplanar graphs can be seen as dissections of a convex polygon, and their numb... |

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Citation Context ...re colored white, and the n-node rooted outerplanar maps. Recall that a graph (or a map) is k-connected if G has more than k nodes and if, for every subset X of fewer than k nodes, G n X is connected =-=[Die00]-=-. biconnected is a synonym for 2-connected. Theorem 2 There is a bijection, computable in linear time, between the (n 1)-node bicolored rooted trees with a white root, all leaves colored in black, and... |

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4 | M.: Generating random outerplanar graphs
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(Show Context)
Citation Context ...3 nor K 4 as a minor. A linear time recognition algorithm has been given by Mitchell [Mit79]. Labeled and unlabeled outerplanar graphs can be randomly generated in O(n 4 log n) space and O(n 2 ) time =-=[BK03]-=- after a preprocessing of O(n 5 ) time. Among graph properties, outerplanar graphs contain trees, have tree-width at most two, and are exactly the graphs of pagenumber one [Bil92]. Recall that a graph... |

2 |
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Citation Context ...ur contribution is an optimal 3n-bit encoding for outerplanar maps. We point out that there exist many 1-page embeddings for a graph of pagenumber one. From the asymptotic formula of Flajolet and Noy =-=[FN99]-=-, any encoding of 1-page embeddings requires 3:37n bits 1 . Let us sketch our technique. First we show that an outerplanar map admits a canonical decomposition into a particular rooted spanning tree (... |

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