Counting outerplanar maps

A map is outerplanar if all its vertices lie in the outer face. We enumerate various classes of rooted outerplanar maps with respect to the number of edges and vertices. The proofs involve several bijections with lattice paths. As a consequence of our results, we obtain an e cient scheme for encoding simple outerplanar maps.Peer ReviewedPostprint (published version

Enumeration of chordal planar graphs and maps

We determine the number of labelled chordal planar graphs with n vertices, which is asymptotically for a constant and . We also determine the number of rooted simple chordal planar maps with n edges, which is asymptotically , where , , and s is an algebraic number of degree 12. The proofs are based on combinatorial decompositions and singularity analysis. Chordal planar graphs (or maps) are a natural example of a subcritical class of graphs in which the class of 3-connected graphs is relatively rich. The 3-connected members are precisely chordal triangulations, those obtained starting from by repeatedly adding vertices adjacent to an existing triangular face.We gratefully acknowledge earlier discussions on this project with Erkan Narmanli. M.N. was supported by grants MTM2017-82166-P and PID2020-113082GB-I00, the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M). C.R. was supported by the grant Beatriu de Pinós BP2019, funded by the H2020 COFUND project No 801370 and AGAUR (the Catalan agency for management of university and research grants), and the grant PID2020-113082GB-I00 of the Spanish Ministry of Science and Innovation.Peer ReviewedPostprint (author's final draft

A lattice on Dyck paths close to the Tamari lattice

We introduce a new poset structure on Dyck paths where the covering relation is a particular case of the relation inducing the Tamari lattice. We prove that the transitive closure of this relation endows Dyck paths with a lattice structure. We provide a trivariate generating function counting the number of Dyck paths with respect to the semilength, the numbers of outgoing and incoming edges in the Hasse diagram. We deduce the numbers of coverings, meet and join irreducible elements. As a byproduct, we present a new involution on Dyck paths that transports the bistatistic of the numbers of outgoing and incoming edges into its reverse. Finally, we give a generating function for the number of intervals, and we compare this number with the number of intervals in the Tamari lattice

Enumeration of chordal planar graphs and maps

We determine the number of labelled chordal planar graphs with $n$ vertices, which is asymptotically $c_1\cdot n^{-5/2} \gamma^n n!$ for a constant $c_1>0$ and $\gamma \approx 11.89235$. We also determine the number of rooted simple chordal planar maps with $n$ edges, which is asymptotically $c_2 n^{-3/2} \delta^n$, where $\delta = 1/\sigma \approx 6.40375$, and $\sigma$ is an algebraic number of degree 12. The proofs are based on combinatorial decompositions and singularity analysis. Chordal planar graphs (or maps) are a natural example of a subcritical class of graphs in which the class of 3-connected graphs is relatively rich. The 3-connected members are precisely chordal triangulations, those obtained starting from $K_4$ by repeatedly adding vertices adjacent to an existing triangular face

Counting outerplanar maps

A map is outerplanar if all its vertices lie in the outer face. We enumerate various classes of rooted outerplanar maps with respect to the number of edges and vertices. The proofs involve several bijections with lattice paths. As a consequence of our results, we obtain an e cient scheme for encoding simple outerplanar maps.Peer Reviewe