Enumeration of chordal planar graphs and maps

02/27/2022

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by   Jordi Castellví, et al.

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We determine the number of labelled chordal planar graphs with n vertices, which is asymptotically c_1· n^-5/2γ^n n! for a constant c_1>0 and γ≈ 11.89235. We also determine the number of rooted simple chordal planar maps with n edges, which is asymptotically c_2 n^-3/2δ^n, where δ = 1/σ≈ 6.40375, and σ 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.