# Double geodetic number of a graph

• Volume: 32, Issue: 1, page 109-119
• ISSN: 2083-5892

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## Abstract

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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.

## How to cite

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A.P. Santhakumaran, and T. Jebaraj. "Double geodetic number of a graph." Discussiones Mathematicae Graph Theory 32.1 (2012): 109-119. <http://eudml.org/doc/270828>.

@article{A2012,
abstract = {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.},
author = {A.P. Santhakumaran, T. Jebaraj},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {geodetic number; weak-extreme vertex; double geodetic set; double geodetic number},
language = {eng},
number = {1},
pages = {109-119},
title = {Double geodetic number of a graph},
url = {http://eudml.org/doc/270828},
volume = {32},
year = {2012},
}

TY - JOUR
AU - A.P. Santhakumaran
AU - T. Jebaraj
TI - Double geodetic number of a graph
JO - Discussiones Mathematicae Graph Theory
PY - 2012
VL - 32
IS - 1
SP - 109
EP - 119
AB - 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.
LA - eng
KW - geodetic number; weak-extreme vertex; double geodetic set; double geodetic number
UR - http://eudml.org/doc/270828
ER -

## References

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1. [1] F. Buckley and F. Harary, Distance in Graphs (Addison-Wesley, Redwood City, CA, 1990). Zbl0688.05017
2. [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. Zbl0987.05047
3. [3] G. Chartrand, F. Harary, H.C. Swart and P. Zhang, Geodomination in graphs, Bulletin ICA 31 (2001) 51-59. Zbl0969.05048
4. [4] F. Harary, Graph Theory (Addision-Wesely, 1969).
5. [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. Zbl0825.68490
6. [6] R. Muntean and P. Zhang, On geodomonation in graphs, Congr. Numer. 143 (2000) 161-174. Zbl0969.05047
7. [7] P.A. Ostrand, Graphs with specified radius and diameter, Discrete Math. 4 (1973) 71-75, doi: 10.1016/0012-365X(73)90116-7. Zbl0265.05123

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