# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

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# Discussiones Mathematicae Graph Theory

## Double Geodetic Number of a Graph

 A.P. Santhakumaran Department of Mathematics St.Xavier's College (Autonomous) Palayamkottai -- 627 002, India T. Jebaraj Department of Mathematics C.S.I. Institute of Technology Thovalai -- 629 302, India

## 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.

Keywords: geodetic number, weak-extreme vertex, double geodetic set, double geodetic number

2010 Mathematics Subject Classification: 05C12.

## References

 [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.