Geodesic
The forcing geodetic number of a graph
Discussiones Mathematicae. Graph Theory, Tome 19 (1999) no. 1, pp. 45-58.

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For two vertices u and v of a graph G, the set I(u, v) consists of all vertices lying on some u-v geodesic in G. If S is a set of vertices of G, then I(S) is the union of all sets I(u,v) for u, v ∈ S. A set S is a geodetic set if I(S) = V(G). A minimum geodetic set is a geodetic set of minimum cardinality and this cardinality is the geodetic number g(G). A subset T of a minimum geodetic set S is called a forcing subset for S if S is the unique minimum geodetic set containing T. The forcing geodetic number f_G(S) of S is the minimum cardinality among the forcing subsets of S, and the forcing geodetic number f(G) of G is the minimum forcing geodetic number among all minimum geodetic sets of G. The forcing geodetic numbers of several classes of graphs are determined. For every graph G, f(G) ≤ g(G). It is shown that for all integers a, b with 0 ≤ a ≤ b, a connected graph G such that f(G) = a and g(G) = b exists if and only if (a,b) ∉ (1,1),(2,2).
Keywords: geodetic set, geodetic number, forcing geodetic number 
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Chartrand, Gary; Zhang, Ping. The forcing geodetic number of a graph. Discussiones Mathematicae. Graph Theory, Tome 19 (1999) no. 1, pp. 45-58. http://geodesic.mathdoc.fr/item/DMGT_1999_19_1_a3/

[1] G. Chartrand, F. Harary and P. Zhang, The geodetic number of a graph, Networks (to appear).

[2] G. Chartrand, F. Harary, and P. Zhang, On the hull number of a graph, Ars Combin. (to appear).