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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1428, author = {Seog-Jin Kim and Douglas B. West}, title = {Triangle-free planar graphs with minimum degree 3 have radius at least 3}, journal = {Discussiones Mathematicae Graph Theory}, volume = {28}, year = {2008}, pages = {563-566}, zbl = {1172.05317}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1428} }
Seog-Jin Kim; Douglas B. West. Triangle-free planar graphs with minimum degree 3 have radius at least 3. Discussiones Mathematicae Graph Theory, Tome 28 (2008) pp. 563-566. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1428/
[000] [1] P. Erdös, J. Pach, R. Pollack and Zs. Tuza, Radius, diameter, and minimum degree, J. Combin. Theory (B) 47 (1989) 73-79, doi: 10.1016/0095-8956(89)90066-X.
[001] [2] J. Harant, An upper bound for the radius of a 3-connected planar graph with bounded faces, Contemporary methods in graph theory (Bibliographisches Inst., Mannheim, 1990), 353-358.
[002] [3] J. Plesník, Critical graphs of given diameter, Acta Fac. Rerum Natur. Univ. Comenian. Math. 30 (1975) 71-93. | Zbl 0318.05115