Geodesic
Scaffold for the polyhedral embedding of cubic graphs
Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 2, pp. 351-384

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Let G be a cubic graph and Π be a polyhedral embedding of this graph. The extended graph, G^e, of Π is the graph whose set of vertices is V(G^e)=V(G) and whose set of edges E(G^e) is equal to E(G) ∪𝒮, where 𝒮 is constructed as follows: given two vertices t_0 and t_3 in V(G^e) we say [t_0 t_3] ∈𝒮, if there is a 3-path, (t_0 t_1 t_2 t_3) ∈ G that is a Π-facial subwalk of the embedding. We prove that there is a one to one correspondence between the set of possible extended graphs of G and polyhedral embeddings of G.
Keywords: cubic graph, polyhedral embedding 
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     title = {Scaffold for the polyhedral embedding of cubic graphs},
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Aguilar-Campos, Flor; Araujo-Pardo, Gabriela; García-Colín, Natalia. Scaffold for the polyhedral embedding of cubic graphs. Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 2, pp. 351-384. http://geodesic.mathdoc.fr/item/DMGT_2023_43_2_a3/