Discussiones Mathematicae Graph Theory 32(3) (2012) 449-459
doi: 10.7151/dmgt.1609

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Fractional Distance Domination in Graphs

S. Arumugam1,2, Varughese Mathew3 and K. Karuppasamy1

1National Centre for Advanced Research in Discrete Mathematics (n-CARDMATH)

Abstract

Let G = (V,E) be a connected graph and let k be a positive integer with k ≤ rad(G). A subset D ⊆ V is called a distance k-dominating set of G if for every v ∈ V −D, there exists a vertex u ∈ D such that d(u,v) ≤ k. In this paper we study the fractional version of distance k-domination and related parameters.

Keywords: domination, distance k-domination, distance k-dominating
function, k-packing, fractional distance k-domination

2010 Mathematics Subject Classification: 05C69, 05C72.

References

[1]S. Arumugam, K. Karuppasamy and I. Sahul Hamid, Fractional global domination in graphs, Discuss. Math. Graph Theory 30 (2010) 33--44, doi: 10.7151/dmgt.1474.
[2]E.J. Cockayne, G. Fricke, S.T. Hedetniemi and C.M. Mynhardt, Properties of minimal dominating functions of graphs, Ars Combin. 41 (1995) 107--115.
[3]G. Chartrand and L. Lesniak, Graphs & Digraphs, Fourth Edition, Chapman & Hall/CRC (2005).
[4]G.S. Domke, S.T. Hedetniemi and R.C. Laskar, Generalized packings and coverings of graphs, Congr. Numer. 62 (1988) 259--270.
[5]G.S. Domke, S.T. Hedetniemi and R.C. Laskar, Fractional packings, coverings, and irredundance in graphs, Congr. Numer. 66 (1988) 227--238.
[6]D.L. Grinstead and P.J. Slater, Fractional domination and fractional packings in graphs, Congr. Numer. 71 (1990) 153--172.
[7]E.O. Hare, k-weight domination and fractional domination of Pm × Pn, Congr. Numer. 78 (1990) 71--80.
[8]J.H. Hattingh, M.A. Henning and J.L. Walters, On the computational complexity of upper distance fractional domination, Australas. J. Combin. 7 (1993) 133--144.
[9]T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998).
[10]T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in graphs: Advanced Topics (Marcel Dekker, New York, 1998).
[11]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).
[12]A. Meir and J.W. Moon, Relations between packing and covering numbers of a tree, Pacific J. Math. 61 (1975) 225--233.
[13]R.R. Rubalcaba, A. Schneider and P.J. Slater, A survey on graphs which have equal domination and closed neighborhood packing numbers, AKCE J. Graphs. Combin. 3 (2006) 93--114.
[14]E.R. Scheinerman and D.H. Ullman, Fractional Graph Theory: A Rational Approach to the Theory of Graphs (John Wiley & Sons, New York, 1997).
[15]D. Vukičević and A. Klobučar, k-dominating sets on linear benzenoids and on the infinite hexagonal grid, Croatica Chemica Acta 80 (2007) 187--191.

Received 22 December 2010
Revised 12 August 2011
Accepted 16 August 2011