# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2017-2018): c. 84%

# Discussiones Mathematicae Graph Theory

## 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.

## References

 [1] B. Bresar, S. Klavžar and A.T. Horvat, On the geodetic number and related metric sets in Cartesian product graphs, Discrete Math. 308 (2008) 5555-5561, doi: 10.1016/j.disc.2007.10.007. [2] F. Buckley and F. Harary, Distance in Graphs (Addison-Wesley, Redwood City, CA, 1990). [3] G. Chartrand, F. Harary and P. Zhang, On the Geodetic Number of a Graph, Networks 39 (2002) 1-6, doi: 10.1002/net.10007. [4] G. Chartrand, F. Harary, H.C. Swart and P. Zhang, Geodomination in Graphs, Bulletin of the ICA 31 (2001) 51-59. [5] G. Chartrand and P. Zhang, Introduction to Graph Theory (Tata McGraw-Hill Edition, New Delhi, 2006). [6] F. Harary, E. Loukakis and C. Tsouros, The geodetic number of a graph, Mathl. Comput. Modeling 17 (1993) 89-95, doi: 10.1016/0895-7177(93)90259-2. [7] F. Harary and T.W. Haynes, Double domination in graphs, Ars Combin. 55 (2000) 201-213. [8] W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition (Wiley-Interscience, New York, 2000). [9] A.P. Santhakumaran and S.V. Ullas Chandran, The hull number of strong product of graphs, (communicated).