Geodesic
Double geodetic number of a graph
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 1, pp. 109-119

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For a connected graph G of order n, a set S of vertices is called a double geodetic set of G if for each pair of vertices x,y in G there exist vertices u,v ∈ S such that x,y ∈ I[u,v]. The double geodetic number dg(G) is the minimum cardinality of a double geodetic set. Any double godetic of cardinality dg(G) is called dg-set of G. The double geodetic numbers of certain standard graphs are obtained. It is shown that for positive integers r,d such that r d ≤ 2r and 3 ≤ a ≤ b there exists a connected graph G with rad G = r, diam G = d, g(G) = a and dg(G) = b. Also, it is proved that for integers n, d ≥ 2 and l such that 3 ≤ k ≤ l ≤ n and n-d-l+1 ≥ 0, there exists a graph G of order n diameter d, g(G) = k and dg(G) = l.
Keywords: geodetic number, weak-extreme vertex, double geodetic set, double geodetic number 
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Santhakumaran, A.; Jebaraj, T. Double geodetic number of a graph. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 1, pp. 109-119. http://geodesic.mathdoc.fr/item/DMGT_2012_32_1_a8/