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Enumeration of discrete structures: algebraic, analytic, probabilistic and algorithmic methods for enriched planar graphs and planar maps

Periodic Report Summary 1 - COUNTGRAPH (Enumeration of discrete structures: algebraic, analytic, probabilistic and algorithmic methods for enriched planar graphs and planar maps)

The CountGraph project fits inside the study of large combinatorial discrete structures, mainly from an enumerative and probabilistic point of view by using the setting of Analytic Combinatorics. The powerful analytic tools emerging from singularity analysis on formal power series, developed mainly by the french school around Philippe Flajolet, provide a general framework to describe the common shape of a (large) discrete object chosen uniformly at random among elements of fixed size. The interest to this area has considerably increased in the last years specially due to its deep connections with theoretical physics (2D quantum gravity and percolation theory), as well as the groundbreaking achievements in the area and the applications of these techniques in the study of the Erdös-Rényi model G(n,p).

More specifically, the CountGraph project aims to study planar maps and planar graphs. Maps (or embedded graphs) have become an important domain in discrete mathematics, combinatorics and probability theory. Generating functions, combined with bijective techniques, have provided a key tool to deal with enumerative questions arising in this field, and the manipulation of these algebraic objects has given a rich collection of results. On the other side, labelled planar graphs are graphs that can be drawn in the sphere without edge crossings. Such objects play a central role in graph theory, and had shown to play a central role in several other areas, for instance discrete mathematics, geometric combinatorics and probability theory. 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 building on 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.

In this direction, we have already obtained significant results that we resume here:

- Exact and asymptotic enumeration of bipartite series-parallel graphs.
- Subgraph statistics in bipartite series-parallel graphs and, more generally, in subcritical graph classes.
- New enumerative and probabilistic results for restricted vertex-degree planar graphs.
- New algorithmic results (by means of parametrized complexity techniques) for the the Plane Diameter Completion problem.

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.

The permanent address of the project is the following one: