Geodesic
On the forcing geodetic and forcing steiner numbers of a graph
Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 4, pp. 611-624

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For a connected graph G = (V,E), a set W ⊆ V is called a Steiner set of G if every vertex of G is contained in a Steiner W-tree of G. The Steiner number s(G) of G is the minimum cardinality of its Steiner sets and any Steiner set of cardinality s(G) is a minimum Steiner set of G. For a minimum Steiner set W of G, a subset T ⊆ W is called a forcing subset for W if W is the unique minimum Steiner set containing T. A forcing subset for W of minimum cardinality is a minimum forcing subset of W. The forcing Steiner number of W, denoted by fₛ(W), is the cardinality of a minimum forcing subset of W. The forcing Steiner number of G, denoted by fₛ(G), is fₛ(G) = minfₛ(W), where the minimum is taken over all minimum Steiner sets W in G. The geodetic number g(G) and the forcing geodetic number f(G) of a graph G are defined in [2]. It is proved in [6] that there is no relationship between the geodetic number and the Steiner number of a graph so that there is no relationship between the forcing geodetic number and the forcing Steiner number of a graph. We give realization results for various possibilities of these four parameters.
Keywords: geodetic number, Steiner number, forcing geodetic number, forcing Steiner number 
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Santhakumaran, A.; John, J. On the forcing geodetic and forcing steiner numbers of a graph. Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 4, pp. 611-624. http://geodesic.mathdoc.fr/item/DMGT_2011_31_4_a0/