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

Voir la notice de l'article provenant de la source Library of Science

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 
@article{DMGT_2012_32_1_a8,
     author = {Santhakumaran, A. and Jebaraj, T.},
     title = {Double geodetic number of a graph},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {109--119},
     publisher = {mathdoc},
     volume = {32},
     number = {1},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2012_32_1_a8/}
}
TY  - JOUR
AU  - Santhakumaran, A.
AU  - Jebaraj, T.
TI  - Double geodetic number of a graph
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2012
SP  - 109
EP  - 119
VL  - 32
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2012_32_1_a8/
LA  - en
ID  - DMGT_2012_32_1_a8
ER  - 
%0 Journal Article
%A Santhakumaran, A.
%A Jebaraj, T.
%T Double geodetic number of a graph
%J Discussiones Mathematicae. Graph Theory
%D 2012
%P 109-119
%V 32
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2012_32_1_a8/
%G en
%F DMGT_2012_32_1_a8
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/

[1] F. Buckley and F. Harary, Distance in Graphs (Addison-Wesley, Redwood City, CA, 1990).

[2] G. Chartrand, F. Harary and P. Zhang, On the geodetic number of a graph, Networks 39 (2002) 1-6, doi: 10.1002/net.10007.

[3] G. Chartrand, F. Harary, H.C. Swart and P. Zhang, Geodomination in graphs, Bulletin ICA 31 (2001) 51-59.

[4] F. Harary, Graph Theory (Addision-Wesely, 1969).

[5] F. Harary, E. Loukakis and C. Tsouros, The geodetic number of a graph, Math. Comput. Modeling 17 (1993) 89-95, doi: 10.1016/0895-7177(93)90259-2.

[6] R. Muntean and P. Zhang, On geodomonation in graphs, Congr. Numer. 143 (2000) 161-174.

[7] P.A. Ostrand, Graphs with specified radius and diameter, Discrete Math. 4 (1973) 71-75, doi: 10.1016/0012-365X(73)90116-7.