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## Proyecciones (Antofagasta)

##
*Print version* ISSN 0716-0917

### Proyecciones (Antofagasta) vol.33 no.1 Antofagasta Mar. 2014

#### http://dx.doi.org/10.4067/S0716-09172014000100003

**Global neighbourhood domination**

**S. V. Siva Rama Raju**

*Ibra College of Technology,** **Sultanate of Oman*

**I. H. Nagaraja Rao**

*G. V. P. P. G. Courses**,** **India *

**ABSTRACT**

*A subset D of vertices of a graph G is called a global neighbourhood dominating set(gnd - set) if D is a dominating set for both G and G ^{N}, where G^{N} is the neighbourhood graph of G. The global neighbourhood domination number(gnd - number) is the minimum cardinality of a global neighbourhood dominating set of G and is denoted by *γ

*γ*

_{gn}(G). In this paper sharp bounds for

_{gn}*,*

*are supplied for graphs whose girth is greater than three. Exact values ofthis number for paths and cycles are presented as well. The characterization result for a subset ofthe vertex set of G to be a global neighbourhood dominating set for G is given and also characterized the graphs of order n having gnd -numbers*1, 2,

*n*—

*1,n*— 2,

*n.*

**Subject Classification :** *05C69.*

**Keywords** : *Global neighbourhood domination, global neighbourhood domination number, global domination, restrained domination, connected domination.*

**REFERENCES**

*[1] Bondy J. A. and Murthy, U. S. R., Graph theory with Applications, The Macmillan Press Ltd, (1976). *

*[2] R. C. Brigham, R. D. Dutton,On Neighbourhood Graphs, J. Combin. inform. System Sci, 12, pp. 75-85, (1987). *

*[3] G. S. Domke, etal., Restrained Domination in Graphs, Discrete Mathematics, 203, pp. 61-69, (1999). *

*[4] T. W. Haynes, S. T. Hedetneimi, P. J. Slater, Fundamentals of Dominations in Graphs Marcel Dekker, New York, (1988). *

*[5] I. H. Naga Raja Rao, S. V. Siva Rama Raju, On Semi-Complete Graphs, International Journal Of Computational Cognition, Vol.7(3), pp. 50-54, (2009). *

*[6] D. F. Rall, Congr. Numer., 80, pp. 89-95, (1991). *

*[7] E. Sampathkumar, H. B. Walikar,The connected Domination Number of a Graph, J. Math. Phy. Sci, Vol.13, pp. 607-613, (1979). *

*[8] E. Sampathkumar,The global domination number of a graph, J. Math.Phy. Sci, Vol. 23 (5), (1989). *

**S. V. Siva Rama Raju**

*Department of Mathematics Ibra College of Technology Ibra,*

*Sultanate of Oman*

*e-mail :shi_vram2006@yahoo.co.in*

**I. H. Nagaraja Rao**

*Department of Mathematics G. V. P. P. G. Courses Visakhapatnam, India*

*e-mail :* *ihnrao@yahoo.com*

*Received : September 2012. Accepted : October 2013*.