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Enumeration of discrete structures: algebraic, analytic, probabilistic and algorithmic methods for enriched planar graphs and planar maps (COUNTGRAPH)
Start date: Apr 1, 2014, End date: Mar 31, 2018 PROJECT  FINISHED 

Our aim in this project is to built on recent combinatorial and algorithmic progress to attack a series of problems that have independently surfaced in the graph enumeration setting, as well as to develop a more systematic approach that works on a wide class of random graph families.The central objects under study are planar graphs and planar embedded graphs (also called maps). The enumeration theory of these objects was initiated by Tutte in the 1960s when studying rooted planar maps; later, in the 1970s, there has been more emphasis on asymptotics and the interplay between graph enumeration and the theory of random graphs. The field has grown enormously since then and many classes of maps have been studied, including maps in arbitrary surfaces. Moreover, deep connections with algebra, low-dimensional topology, probability and statistical physics have been uncovered.Recently the interest in planar maps and graphs has considerably increased, due to fundamental constructions by Schaeffer (bijections for planar maps in terms of enriched tree structures), and Giménez and Noy (generating function techniques joint with analytic tools). Our objective is to continue the lines of these achievements and explore their interactions with other domains, specially with computer science.More precisely, the main goals of this project are to develop new tools to deal with open questions in the field, including the study of bipartite families of graphs, unlabelled families of graphs, and planar graphs with restricted vertex degrees, among other questions. In most of the cases, the interaction between the map enumeration domain and the algorithmic setting will be strongly explored.The main techniques exploited in this project arise from the Analytic Combinatorics setting: that is, the combinatorial structure is translated into equations of generating functions, that can be studied by means of complex analytic methods, joint with probabilistic techniques.

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