Discussiones Mathematicae Graph Theory 30(1) (2010)
33-44
doi: 10.7151/dmgt.1474
Subramanian Arumugam, Kalimuthu Karuppasamy
Core Group Research Facility (CGRF) |
Ismail Sahul Hamid
Department of Mathematics |
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.
[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