On the forcing connected geodetic number and the connected geodetic number of a graph

For two vertices u and v of a nontrivial connected graph G, the set I[u, v] consists of all vertices lying on some u - v geodesic in G, including u and v. For S subset of 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 subset of V(G) is a connected geodetic set of G if I[S] = V(G) and the subgraph in G induced by S is connected. The minimum cardinality of a connected geodetic set of G is the connected geodetic number g(c)(G) of G and a connected geodetic set of G whose cardinality equals g(c)(G) is a minimum connected geodetic set of G. A subset T of a minimum connected geodetic set S is a forcing subset for S if S is the unique minimum connected geodetic set of G containing T. The forcing connected geodetic number f(c)(S) of S is the minimum cardinality of a forcing subset of S and the forcing connected geodetic number f(c)(G) of G is the minimum forcing connected geodetic number among all minimum connected geodetic sets of G. Therefore, 0 <= f(c)(G) <= g(c)(G). We determine all pairs a, b of integers such that f(c)(G) = a and g(c)(G) = b for some nontrivial connected graph G. We also consider a problem of realizable triples of integers.