A Complete Grammar for Decomposing a Family of Graphs into 3-connected Components

Tutte has described in the book "Connectivity in graphs" a canonical decomposition of any graph into 3-connected components. In this article we translate (using the language of symbolic combinatorics) Tutte's decomposition into a general grammar expressing any family of graphs (with some stability conditions) in terms of the 3-connected subfamily. A key ingredient we use is an extension of the so-called dissymmetry theorem, which yields negative signs in the grammar. As a main application we recover in a purely combinatorial way the analytic expression found by Gim\'enez and Noy for the series counting labelled planar graphs (such an expression is crucial to do asymptotic enumeration and to obtain limit laws of various parameters on random planar graphs). Besides the grammar, an important ingredient of our method is a recent bijective construction of planar maps by Bouttier, Di Francesco and Guitter.Comment: 39 page

Dynamic programming for graphs on surfaces

We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs on n vertices and branchwidth at most k. Our technique applies to general families of problems where standard dynamic programming runs in 2O(k·log k). Our approach combines tools from topological graph theory and analytic combinatorics.Postprint (updated version

Statistics of planar graphs viewed from a vertex: A study via labeled trees

We study the statistics of edges and vertices in the vicinity of a reference vertex (origin) within random planar quadrangulations and Eulerian triangulations. Exact generating functions are obtained for theses graphs with fixed numbers of edges and vertices at given geodesic distances from the origin. Our analysis relies on bijections with labeled trees, in which the labels encode the information on the geodesic distance from the origin. In the case of infinitely large graphs, we give in particular explicit formulas for the probabilities that the origin have given numbers of neighboring edges and/or vertices, as well as explicit values for the corresponding moments.Comment: 36 pages, 15 figures, tex, harvmac, eps

Census of Planar Maps: From the One-Matrix Model Solution to a Combinatorial Proof

We consider the problem of enumeration of planar maps and revisit its one-matrix model solution in the light of recent combinatorial techniques involving conjugated trees. We adapt and generalize these techniques so as to give an alternative and purely combinatorial solution to the problem of counting arbitrary planar maps with prescribed vertex degrees.Comment: 29 pages, 14 figures, tex, harvmac, eps

Planar maps and continued fractions

We present an unexpected connection between two map enumeration problems. The first one consists in counting planar maps with a boundary of prescribed length. The second one consists in counting planar maps with two points at a prescribed distance. We show that, in the general class of maps with controlled face degrees, the solution for both problems is actually encoded into the same quantity, respectively via its power series expansion and its continued fraction expansion. We then use known techniques for tackling the first problem in order to solve the second. This novel viewpoint provides a constructive approach for computing the so-called distance-dependent two-point function of general planar maps. We prove and extend some previously predicted exact formulas, which we identify in terms of particular Schur functions.Comment: 47 pages, 17 figures, final version (very minor changes since v2