## An Information-Theoretic Upper Bound of Planar Graphs Using Triangulation (2003)

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Citations: | 21 - 4 self |

### BibTeX

@MISC{Bonichon03aninformation-theoretic,

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

title = {An Information-Theoretic Upper Bound of Planar Graphs Using Triangulation},

year = {2003}

}

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### Abstract

We propose a new linear time algorithm to represent a planar graph. Based on a specific triangulation of the graph, our coding takes on average 5.03 bits per node, and 3.37 bits per node if the graph is maximal. We derive from this representation that the number of unlabeled planar graphs with n nodes is at most 2 n+O(log n) where 5.007. The current lower bound is 2 n+(log n) for 4.71. We also show that almost all unlabeled and almost all labeled n-node planar graphs have at least 1.70n edges and at most 2.54n edges.

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Citation Context ...nstruction for planar graphs can be used for Networking, in particular to improve the result of [26]. 1.3 Outline of the paper Let us sketch our technique (due to space limitation the proofs moved to =-=[8]-=-). A natural approach to represent an n-node planar graph G is to consider a triangulation of G, i.e., a supergraph S of G such that S is planar, has n nodes 1 This result appears in the full version.... |

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2 |
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Citation Context ... Note that enumeration of simple planar maps gives an upper bound on planar graphs whose algebraic generating function has been given by Liu [23]. Their number is asymptotic to 2 5:098n+O(log n) (cf. =-=[9]-=-), providing an upper bound for unlabeled and labeled planar graphs. 1.2 Our results In this paper we show an upper bound of 2 5:007n+O(log n) on p(n), the number of unlabeled planar graphs with n nod... |

2 |
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Citation Context ...ore than n!2 5:22n+o(n) labeled planar graphs. They also showed that almost all labeled planar graphs have at most 2:56n edges. A lower bound of 13n=7 1:85n has been obtained by Gerke and McDiarmid [=-=18]-=-, improving the 1:5n lower bound of [13]. They also derived that almost all unlabeled planar graphs have at most 2:69n edges. Note that there is no evidence that labeled and unlabeled planar graphs ha... |

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Citation Context ...led 2-connected planar graphs tends to n!2 4:71n+O(log n) . Note that enumeration of simple planar maps gives an upper bound on planar graphs whose algebraic generating function has been given by Liu =-=[23]-=-. Their number is asymptotic to 2 5:098n+O(log n) (cf. [9]), providing an upper bound for unlabeled and labeled planar graphs. 1.2 Our results In this paper we show an upper bound of 2 5:007n+O(log n)... |