Geodetic connectivity of graphs

A graph
G
is said to be
n
-geodetically connected if and only if
G
is connected and the removal of at least
n
points is required to increase the distance between any pair of points. "Geodetic" analogs of results such as Menger's theorem and Dirac's "fan" theorem are shown to hold. Some other characterizations of
n
-geodetically connected graphs are obtained, one of which shows geodetic connectivity to be a local property in contrast to the usual connectivity.