# Double geodetic number of a graph

A.P. Santhakumaran; T. Jebaraj

Discussiones Mathematicae Graph Theory (2012)

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

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topA.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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- [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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