Some special minimum kk-geodetically connected graphs

A connected graph GG is kk-geodetically connected (kk-GC) if the removal of less than kk vertices does not affect the distances (lengths of the shortest paths) between any pair of the remaining vertices. As such graphs have important applications in robust system designs, we are interested in the minimum number of edges required to make a kk-GC graph of order nn, and characterizing those minimum kk-GC graphs. When 3