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New bijective links on planar maps

Abstract : This article describes new bijective links on planar maps, which are of incremental complexity and present original features. The first two bijections $\Phi _{1,2}$ are correspondences on oriented planar maps. They can be considered as variations on the classical edge-poset construction for bipolar orientations on graphs, suitably adapted so as to operate only on the embeddings in a simple local way. In turn, $\Phi_{1,2}$ yield two new bijections $F_{1,2}$ between families of (rooted) maps. (i) By identifying maps with specific constrained orientations, $\Phi_2 \circ \Phi_1$ specialises to a bijection $F_1$ between 2-connected maps and irreducible triangulations; (ii) $F_1$ gives rise to a bijection $F_2$ between loopless maps and triangulations, observing that these decompose respectively into 2-connected maps and into irreducible triangulations in a parallel way.
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Eric Fusy. New bijective links on planar maps. 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), 2008, Viña del Mar, Chile. pp.153-166, ⟨10.46298/dmtcs.3628⟩. ⟨hal-01185163⟩

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