Geodesic
Characterization of some extremal graphs with a diameter at most three
Diskretnaya Matematika, Tome 9 (1997) no. 1, pp. 134-146

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We consider the problem to find the lower bound for the number of edges for graphs in which after removing an arbitrary vertex or an arbitrary edge the diameter of the graph obtained does not exceed three. Also, the graphs for which the lower bound determined is attained are enumerated. 
@article{DM_1997_9_1_a10,
     author = {D. L. Belotserkovskii},
     title = {Characterization of some extremal graphs with a diameter at most three},
     journal = {Diskretnaya Matematika},
     pages = {134--146},
     publisher = {mathdoc},
     volume = {9},
     number = {1},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_1997_9_1_a10/}
}
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D. L. Belotserkovskii. Characterization of some extremal graphs with a diameter at most three. Diskretnaya Matematika, Tome 9 (1997) no. 1, pp. 134-146. http://geodesic.mathdoc.fr/item/DM_1997_9_1_a10/