Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the minimal Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a byproduct we derive: (i) a simple interpretation of the formula for the number of plane triangulations with n vertices, (ii) a linear random sampling algorithm, (iii) an explicit and simple information theory optimal encoding. 1

Abstract. This article introduces new algorithms for the uniform random generation of labelled planar graphs. Its principles rely on Boltzmann samplers, as recently developed by Duchon, Flajolet, Louchard, and Schaeffer. It combines the Boltzmann framework, a suitable use of rejection, a new combinatorial bijection found by Fusy, Poulalhon and Schaeffer, as well as a precise analytic description of the generating functions counting planar graphs, which was recently obtained by Giménez and Noy. This gives rise to an extremely efficient algorithm for the random generation of planar graphs. There is a preprocessing step of some fixed small cost; and the expected time complexity of generation is quadratic for exact-size uniform sampling and linear for approximate-size sampling. This greatly improves on the best previously known time complexity for exact-size uniform sampling of planar graphs with n vertices, which was a little over O(n 7). This is the extended and revised journal version of a conference paper with the title “Quadratic exact-size and linear approximate-size random generation of planar graphs”, which appeared in the Proceedings of the International Conference on Analysis of Algorithms (AofA’05), 6-10 June 2005, Barcelona. 1.

We present uniformly available simple enumerative formulae for unrooted planar n-edge maps (counted up to orientation-preserving isomorphism) of numerous classes including arbitrary, loopless, non-separable, eulerian maps and plane trees. All the formulae conform to a certain pattern with respect to the terms of the sum over t j n; t! n: Namely, these terms, which correspond to non-trivial automorphisms of the maps, prove to be of the form OE \Gamma

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Abstract. We solve three enumerative problems concerning families of planar maps. More precisely, we establish algebraic equations for the generating function of non-separable triangulations in which all vertices have degree at least d, for a certain value d chosen in {3, 4, 5}. The originality of the problem lies in the fact that degree restrictions are placed both on vertices and faces. Our proofs first follow Tutte’s classical approach: we decompose maps by deleting the root and translate the decomposition into an equation satisfied by the generating function of the maps under consideration. Then we proceed to solve the equation obtained using a recent technique that extends the so-called quadratic method. 1.

Abstract. For each positive integer d, we present a bijection between the set of planar maps of girth d inside a d-gon and a set of decorated plane trees. The bijection has the property that each face of degree k in the map corresponds to a vertex of degree k in the tree, so that maps of girth d can be counted according to the degree distribution of their faces. More precisely, we obtain for each integer d an explicit expression for the multivariate series Fd(xd,xd+1,xd+2,...) counting rooted maps of girth d inside a d-gon, where each variable xk marks the number of inner faces of degree k. The series F1 (corresponding to maps inside a loop) was already computed bijectively by Bouttier, Di Francesco and Guitter, but for d ≥ 2 the expression of Fd is new. As special cases, we recover several known bijections (bipartite maps, loopless triangulations, simple triangulations, simple quadrangulations, etc.). Our strategy is based on the use of a “master bijection”, introduced by the authors inaprevious paper, between aclass oforiented planar mapsand a class of decorated trees. We obtain our bijections for maps of girth d by specializing the master bijection. Indeed, by defining some “canonical orientations ” for maps of girth d, it is possible to identify the class of maps of girth d inside a d-gon with a class of oriented maps on which the master bijection specializes nicely. The same strategy was already used in a previous article in order to count d-angulations of girth d, and what we present here is a very significant extension of those results. 1.

This article introduces a new algorithm for the random generation of labelled planar graphs. Its principles rely on Boltzmann samplers, as recently developed by Duchon, Flajolet, Louchard, and Schaeffer. It combines the Boltzmann framework, a suitable use of rejection, a new combinatorial bijection found by Fusy, Poulalhon and Schaeffer, as well as a precise analytic description of the generating functions counting planar graphs, which was recently obtained by Giménez and Noy. This gives rise to an extremely efficient algorithm for the random generation of planar graphs. There is a preprocessing step of some fixed small cost. Then, the expected time complexity of generation is quadratic for exact-size uniform sampling and linear for approximate-size sampling. This greatly improves on the best previously known time complexity for exact-size uniform sampling of planar graphs with n vertices, which was a little over O(n 7).