The upper forcing geodetic number of a graph

For vertices u and v in a nontrivial connected graph G, the closed interval I[u, v] consists of u, v, and all vertices lying in some u-v geodesic of G. For S subset of or equal to V(G), the set I[S] is the union of all sets I[u, v] for u, v is an element of S. A set S of vertices of a graph G is a geodetic set in G if I[S] = V(G). The minimum cardinality of a geodetic set in G is its geodetic number g(G). A subset T of a minimum geodetic set S in a graph G is a forcing subset for S if S is the unique minimum geodetic set containing T. The forcing geodetic number f (S) of S in G is the minimum cardinality of a forcing subset for S, and the upper forcing geodetic number f(+) (G) of the graph G is the maximum forcing geodetic number among all minimum geodetic sets of G. Thus 0 less than or equal to f(+) (G) less than or equal to g (G) for every graph G. The upper forcing geodetic numbers of several classes of graphs are determined. It is shown that for every pair a, b of integers with 0 less than or equal to a less than or equal to b and b greater than or equal to 1, there exists a connected graph G with f(+)(G) = a and g (G) = b if and only if (a, b) is an element of {(1, 1), (2, 2)}.