Geodesic
Biregular (and regular) planar cages
Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 4, pp. 1171-1194

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We study the Cage Problem for biregular planar graphs. This problem has being widely studied for biregular graphs (without the planarity hypothesis). An ({r,m};g)-graph is a biregular graph whose vertices have degrees r and m, for 2≤ r lt; m, and girth g. An ({r,m};g)-cage is an ({r,m};g)-graph of minimum order. In this paper, we determine the triplets of values ( { r, m } ; g) for which there exist planar ({ r, m } ; g)-graphs and for all values we construct examples. Furthermore, we bound the order of the ( { r, m } ; g)-cages and in many instances we build examples that reach the bounds.
Keywords: cages, biregular cages, planar graphs 
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     title = {Biregular (and regular) planar cages},
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Araujo-Pardo, Gabriela; Barrera-Cruz, Fidel; García-Colín, Natalia. Biregular (and regular) planar cages. Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 4, pp. 1171-1194. http://geodesic.mathdoc.fr/item/DMGT_2023_43_4_a15/