A few words about maps
Abstract
In this paper, we survey some properties, encoding, and bijections involving combinatorial maps, double occurrence words, and chord diagrams. We particularly study quasi-trees from a purely combinatorial point of view and derive a topological representation of maps with a given spanning quasi-tree using two fundamental polygons, which extends the representation of planar maps based on the equivalence with bipartite circle graphs. Then, we focus on Depth-First Search trees and their connection with a poset we define on the spanning quasi-trees of a map. We apply the bijections obtained in the first section to the problem of enumerating loopless rooted maps. Finally, we return to the planar case and discuss a decomposition of planar rooted loopless maps and its consequences on planar rooted loopless map enumeration.
Cite
@article{arxiv.2211.08259,
title = {A few words about maps},
author = {Robert Cori and Yiting Jiang and Patrice Ossona de Mendez and Pierre Rosenstiehl},
journal= {arXiv preprint arXiv:2211.08259},
year = {2022}
}
Comments
accepted in the special issue of the European Journal of Combinatorics dedicated to Pierre Rosenstiehl