Discussiones Mathematicae Graph Theory 30(1) (2010) 55-73
doi: 10.7151/dmgt.1476

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THE EDGE GEODETIC NUMBER AND CARTESIAN PRODUCT OF 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 a nontrivial connected graph G = (V(G),E(G)), a set S⊆ V(G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g1(G) of G is the minimum order of its edge geodetic sets. Bounds for the edge geodetic number of Cartesian product graphs are proved and improved upper bounds are determined for a special class of graphs. Exact values of the edge geodetic number of Cartesian product are obtained for several classes of graphs. Also we obtain a necessary condition of G for which g1(G[¯] K2) = g1(G).

Keywords: geodetic number, edge geodetic number, linear edge geodetic set, perfect edge geodetic set, (edge, vertex)-geodetic set, superior edge geodetic set.

2010 Mathematics Subject Classification: 05C12.

References

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Received 28 May 2008
Revised 11 December 2008
Accepted 21 January 2009