Geodesic
On Minimal Geodetic Domination in Graphs
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 3, pp. 403-418

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Let G be a connected graph. For two vertices u and v in G, a u-v geodesic is any shortest path joining u and v. The closed geodetic interval I_G[u, v] consists of all vertices of G lying on any u-v geodesic. For S ⊆ V (G), S is a geodetic set in G if ⋃_u,v ∈ S I_G [u, v] = V (G). Vertices u and v of G are neighbors if u and v are adjacent. The closed neighborhood N_G[v] of vertex v consists of v and all neighbors of v. For S ⊆ V (G), S is a dominating set in G if ⋃_u ∈ S N_G[u] = V (G). A geodetic dominating set in G is any geodetic set in G which is at the same time a dominating set in G. A geodetic dominating set in G is a minimal geodetic dominating set if it does not have a proper subset which is itself a geodetic dominating set in G. The maximum cardinality of a minimal geodetic dominating set in G is the upper geodetic domination number of G. This paper initiates the study of minimal geodetic dominating sets and upper geodetic domination numbers of connected graphs.
Keywords: minimal geodetic dominating set, upper geodetic domination number 
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Nuenay, Hearty M.; Jamil, Ferdinand P. On Minimal Geodetic Domination in Graphs. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 3, pp. 403-418. http://geodesic.mathdoc.fr/item/DMGT_2015_35_3_a0/