Geodesic
Triangle-free planar graphs with minimum degree 3 have radius at least 3
Discussiones Mathematicae. Graph Theory, Tome 28 (2008) no. 3, pp. 563-566

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We prove that every triangle-free planar graph with minimum degree 3 has radius at least 3; equivalently, no vertex neighborhood is a dominating set.
Keywords: planar graph, radius, minimum degree, triangle-free, dominating set 
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Kim, Seog-Jim; West, Douglas. Triangle-free planar graphs with minimum degree 3 have radius at least 3. Discussiones Mathematicae. Graph Theory, Tome 28 (2008) no. 3, pp. 563-566. http://geodesic.mathdoc.fr/item/DMGT_2008_28_3_a14/