Discussiones Mathematicae Graph Theory 32(1) (2012) 109-119
doi: 10.7151/dmgt.1589

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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.

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Received 30 June 2010
Revised 17 January 2011
Accepted 1 February 2011