Geodesic
Extremal embedded graphs
Ars Mathematica Contemporanea, Tome 17 (2019) no. 2, pp. 637-652.

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Let G be a ribbon graph and μ(G) be the number of components of the virtual link formed from G as a cellularly embedded graph via the medial construction. In this paper we first prove that μ(G) ≤ f(G) + γ(G), where f(G) and γ(G) are the number of boundary components and Euler genus of G, respectively. A ribbon graph is said to be extremal if μ(G) = f(G) + γ(G). We then obtain that a ribbon graph is extremal if and only if its Petrial is plane. We introduce a notion of extremal minor and provide an excluded extremal minor characterization for extremal ribbon graphs. We also point out that a related result in the monograph by Ellis-Monaghan and Moffatt is not correct and prove that two related conjectures raised by Huggett and Tawfik hold for more general ribbon graphs.
DOI : 10.26493/1855-3974.1474.d54
Keywords: Ribbon graph, medial graph, Petrie dual, extremal minor, orientation 
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Qi Yan; Xian'an Jin. Extremal embedded graphs. Ars Mathematica Contemporanea, Tome 17 (2019) no. 2, pp. 637-652. doi : 10.26493/1855-3974.1474.d54. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.1474.d54/

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