DMGT

Discussiones Mathematicae Graph Theory 30(1) (2010) 33-44
DOI: https://doi.org/10.7151/dmgt.1474

FRACTIONAL GLOBAL DOMINATION IN GRAPHS

Subramanian Arumugam, Kalimuthu Karuppasamy

Core Group Research Facility (CGRF)
National Centre for Advanced Research in Discrete Mathematics
(n-CARDMATH), Kalasalingam University
Anand Nagar, Krishnankoil-626 190, India
e-mail: s.arumugam.klu@gmail.com
k_karuppasamy@yahoo.co.in

Ismail Sahul Hamid

Department of Mathematics
The Madura College, Madurai-625 011, India
e-mail: sahulmat@yahoo.co.in

Abstract

Let G = (V,E) be a graph. A function g:V→ [0,1] is called a global dominating function (GDF) of G, if for every v ∈ V, g(N[v]) = ∑_{u ∈ N[v]}g(u) ≥ 1 and g([`N(v)]) = ∑_{u ∉ N(v)}g(u) ≥ 1. A GDF g of a graph G is called minimal (MGDF) if for all functions f:V→ [0,1] such that f ≤ g and f(v) ≠ g(v) for at least one v ∈ V, f is not a GDF. The fractional global domination number γ_fg(G) is defined as follows: γ_fg(G) = min{|g|:g is an MGDF of G } where |g| = ∑_{v ∈ V} g(v). In this paper we initiate a study of this parameter.

Keywords: domination, global domination, dominating function, global dominating function, fractional global domination number.

2010 Mathematics Subject Classification: 05C69.

References

[1]	S. Arumugam and R. Kala, A note on global domination in graphs, Ars Combin. 93 (2009) 175-180.
[2]	S. Arumugam and K. Rejikumar, Basic minimal dominating functions, Utilitas Mathematica 77 (2008) 235-247.
[3]	G. Chartrand and L. Lesniak, Graphs & Digraphs (Fourth Edition, Chapman & Hall/CRC, 2005).
[4]	E.J. Cockayne, G. MacGillivray and C.M. Mynhardt, Convexity of minimal dominating funcitons of trees-II, Discrete Math. 125 (1994) 137-146, doi: 10.1016/0012-365X(94)90154-6.
[5]	E.J. Cockayne, C.M. Mynhardt and B. Yu, Universal minimal total dominating functions in graphs, Networks 24 (1994) 83-90, doi: 10.1002/net.3230240205.
[6]	T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., 1998).
[7]	T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs: Advanced Topics (Marcel Dekker, Inc., 1998).
[8]	S.M. Hedetniemi, S.T. Hedetniemi and T.V. Wimer, Linear time resource allocation algorithms for trees, Technical report URI -014, Department of Mathematics, Clemson University (1987).
[9]	E. Sampathkumar, The global domination number of a graph, J. Math. Phys. Sci. 23 (1989) 377-385.
[10]	E.R. Scheinerman and D.H. Ullman, Fractional Graph Theory: A Rational Approch to the Theory of Graphs (John Wiley & Sons, New York, 1997).

Received 19 September 2008
Revised 12 January 2009
Accepted 12 January 2009