Geodesic
On short cycles through prescribed vertices of a polyhedral graph
Discussiones Mathematicae. Graph Theory, Tome 25 (2005) no. 3, pp. 419-426

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Guaranteed upper bounds on the length of a shortest cycle through k ≤ 5 prescribed vertices of a polyhedral graph or plane triangulation are proved.
Keywords: polyhedral graph, triangulation, short cycle, prescribed vertices 
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Hexel, Erhard. On short cycles through prescribed vertices of a polyhedral graph. Discussiones Mathematicae. Graph Theory, Tome 25 (2005) no. 3, pp. 419-426. http://geodesic.mathdoc.fr/item/DMGT_2005_25_3_a17/