A Generic Method for Bijections between Blossoming Trees and Planar Maps

Abstract

This article presents a unified bijective scheme between planar maps and blossoming trees, where a blossoming tree is defined as a spanning tree of the map decorated with some dangling half-edges that enable to reconstruct its faces. Our method generalizes a previous construction of Bernardi by loosening its conditions of application so as to include annular maps, that is maps embedded in the plane with a root face different from the outer face.The bijective construction presented here relies deeply on the theory of $\alpha$-orientations introduced by Felsner, and in particular on the existence of minimal and accessible orientations. Since most of the families of maps can be characterized by such orientations, our generic bijective method is proved to capture as special cases many previously known bijections involving blossoming trees: for example Eulerian maps, $m$-Eulerian maps, non-separable maps and simple triangulations and quadrangulations of a $k$-gon. Moreover, it also permits to obtain new bijective constructions for bipolar orientations and $d$-angulations of girth $d$ of a $k$-gon.As for applications, each specialization of the construction translates into enumerative by-products, either via a closed formula or via a recursive computational scheme. Besides, for every family of maps described in the paper, the construction can be implemented in linear time. It yields thus an effective way to encode or sample planar maps.In a recent work, Bernardi and Fusy introduced another unified bijective scheme; we adopt here a different strategy which allows us to capture different bijections. These two approaches should be seen as two complementary ways of unifying bijections between planar maps and decorated trees.