Geodesic
Stable embeddings on closed surfaces with respect to the minimum length
Naoki Mochizuki1 ; Seiya Negami2 ; Naoki Mochizuki ; Seiya Negami
1Graduate School of Environment and Information Sciences, Yokohama National University, Yokohama, Japan
2Faculty of Environment and Information Sciences, Yokohama National University, Japan
Acta mathematica Universitatis Comenianae, Tome 88 (2019) no. 2, pp. 293-304 
Naoki Mochizuki; Seiya Negami; Naoki Mochizuki; Seiya Negami. Stable embeddings on closed surfaces with respect to the minimum length. Acta mathematica Universitatis Comenianae, Tome 88 (2019) no. 2, pp. 293-304. http://geodesic.mathdoc.fr/item/AMUC_2019_88_2_a9/
@article{AMUC_2019_88_2_a9,
     author = {Naoki Mochizuki and Seiya Negami and Naoki Mochizuki and Seiya Negami},
     title = { Stable embeddings on closed surfaces with respect to the minimum length},
     journal = {Acta mathematica Universitatis Comenianae},
     pages = {293--304},
     year = {2019},
     volume = {88},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/AMUC_2019_88_2_a9/}
}
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Voir la notice de l'article provenant de la source Comenius University

An embedding of a graph on a closed surface with suitable metric is said to be minimum-length embedding if the total sum of lengths of its edges measured by the metric is the minimum among all embeddings isotopic to it and is said to be stable with respect to minimum length if the limit of any convergent sequence of minimum-length embeddings isotopic to it is an embedding of the graph. We shall discuss these notions and shall decide which 4-regular quadrangulations and which 6-regular triangulations on the torus have minimum-length embeddings and are stable with respect to minimum length.