Discussiones Mathematicae Graph Theory 30(4) (2010) 687-700
doi: 10.7151/dmgt.1523

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THE GEODETIC NUMBER OF STRONG PRODUCT GRAPHS

A.P. Santhakumaran and S.V. Ullas Chandran

Department of Mathematics
St. Xavier's College (Autonomous)
Palayamkottai - 627 002, India
e-mail: apskumar1953@yahoo.co.in
e-mail: ullaschandra01@yahoo.co.in

Abstract

For two vertices u and v of a connected graph G, the set IG[u,v] consists of all those vertices lying on u−v geodesics in G. Given a set S of vertices of G, the union of all sets IG[u,v] for u,v ∈ S is denoted by IG[S]. A set S ⊆ V(G) is a geodetic set if IG[S] = V(G) and the minimum cardinality of a geodetic set is its geodetic number g(G) of G. Bounds for the geodetic number of strong product graphs are obtainted and for several classes improved bounds and exact values are obtained.

Keywords: geodetic number, extreme vertex, extreme geodesic graph, open geodetic number, double domination number.

2010 Mathematics Subject Classification: 05C12.

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Received 29 October 2009
Revised 27 February 2010
Accepted 10 March 2010