Discussiones Mathematicae Graph Theory 30(2) (2010) 223-235
doi: 10.7151/dmgt.1488

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ON LOCATING-DOMINATION IN GRAPHS

Mustapha Chellali,  Malika Mimouni

LAMDA-RO Laboratory
Department of Mathematics, University of Blida
B.P. 270, Blida, Algeria
e-mail: m_chellali@yahoo.com

Peter J. Slater

Department of Mathematics and Computer Science Department
University of Alabama in Huntsville
Huntsville, AL 35899 USA
e-mail: slaterp@email.uah.edu

Abstract

A set D of vertices in a graph G = (V,E) is a locating-dominating set (LDS) if for every two vertices u,v of V-D the sets N(u)∩D and N(v)∩D are non-empty and different. The locating-domination number γL(G) is the minimum cardinality of a LDS of G, and the upper locating-domination number, ΓL(G) is the maximum cardinality of a minimal LDS of G. We present different bounds on ΓL(G) and γL(G).

Keywords: upper locating-domination number, locating-domination number.

2010 Mathematics Subject Classification: 05C69.

References

[1] M. Blidia, M. Chellali and O. Favaron, Independence and 2-domination in trees, Australasian J. Combin. 33 (2005) 317-327.
[2] M. Blidia, M. Chellali, O. Favaron and N. Meddah, On k-independence in graphs with emphasis on trees, Discrete Math. 307 (2007) 2209-2216, doi: 10.1016/j.disc.2006.11.007.
[3] M. Blidia, M. Chellali, R. Lounes and F. Maffray, Characterizations of trees with unique minimum locating-dominating sets, submitted.
[4] M. Blidia, M. Chellali, F. Maffray, J. Moncel and A. Semri, Locating-domination and identifying codes in trees, Australasian J. Combin. 39 (2007) 219-232.
[5] M. Blidia, O. Favaron and R. Lounes, Locating-domination, 2-domination and independence in trees, Australasian J. Combin. 42 (2008) 309-316.
[6] M. Farber, Domination, independent domination and duality in strongly chordal graphs, Discrete Appl. Math. 7 (1984) 115-130, doi: 10.1016/0166-218X(84)90061-1.
[7] J.F. Fink, M.S. Jacobson, L.F. Kinch and J. Roberts, On graphs having domination number half their order, Period. Math. Hungar. 16 (1985) 287-293, doi: 10.1007/BF01848079.
[8] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998).
[9] T.W. Haynes, S.T. Hedetniemi and P.J. Slater (eds), Domination in Graphs: Advanced Topics (Marcel Dekker, New York, 1998).
[10] C. Payan and N.H. Xuong, Domination-balanced graphs, J. Graph Theory 6 (1982) 23-32, doi: 10.1002/jgt.3190060104.
[11] G. Ravindra, Well covered graphs, J. Combin. Inform. System. Sci. 2 (1977) 20-21.
[12] P.J. Slater, Domination and location in acyclic graphs, Networks 17 (1987) 55-64, doi: 10.1002/net.3230170105.
[13] P.J. Slater, Dominating and reference sets in graphs, J. Math. Phys. Sci. 22 (1988) 445-455.

Received 16 December 2008
Revised 8 June 2009
Accepted 8 June 2009