Abstract
A three-valued function f: V → {−1, 0, 1} defined on the vertices of a graph G= (V, E) is a minus total dominating function (MTDF) if the sum of its function values over any open neighborhood is at least one. That is, for every υ ∈ V, f(N(υ)) ⩾ 1, where N(υ) consists of every vertex adjacent to υ. The weight of an MTDF is f(V) = Σf(υ), over all vertices υ ∈ V. The minus total domination number of a graph G, denoted γ − t (G), equals the minimum weight of an MTDF of G. In this paper, we discuss some properties of minus total domination on a graph G and obtain a few lower bounds for γ − t (G).
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References
R.B. Allan, R.C. Laskar and S.T. Hedetniemi: A note on total domination. Discrete Math. 49 (1984), 7–13.
D. Archdeacon, J. Ellis-Monaghan, and D. Fisher et al.: Some remarks on domination. Journal of Graph Theory 46 (2004), 207–210.
S. Arumugam and A. Thuraiswamy: Total domination in graphs. Ars Combin. 43 (1996), 89–92.
E. J. Cockayne, R. M. Dawes and S. T. Hedetniemi: Total domination in graphs. Networks 10 (1980), 211–219.
J. E. Dunbar, W. Goddard, S. T. Hedetniemi, M. A. Henning and A. A. McRae: The algorithmic complexity of minus domination in graphs. Discrete Appl. Math. 68 (1996), 73–84.
J. E. Dunbar, S. T. Hedetniemi, M. A. Henning and A. A. McRae: Minus domination in regular graphs. Discrete Math. 149 (1996), 311–312.
J. E. Dunbar, S. T. Hedetniemi, M. A. Henning and A. A. McRae: Minus domination in graphs. Discrete Math. 199 (1999), 35–47.
O. Favaron, M. A. Henning, and C. M. Mynhart et al.: Total domination in graphs with minimum degree three. Journal of Graph Theory 32 (1999), 303–310.
L. Harris and J. H. Hattingh: The algorithmic complexity of certain functional variations of total domination in graphs. Australas. Journal of Combin. 29 (2004), 143–156.
T. W. Haynes, S. T. Hedetniemi and P. J. Slater: Fundamentals of domination in graphs. New York, Marcel Dekker, 1998.
M. A. Henning: Graphs with large total domination number. Journal of Graph Theory 35 (2000), 21–45.
M. A. Henning: Signed total domination in graphs. Discrete Math. 278 (2004), 109–125.
L. Y. Kang, H. K. Kim and M. Y. Sohn: Minus domination number in k-partite graphs. Discrete Math. 227 (2004), 295–300.
H. M. Xing, L. Sun and X. G. Chen: On signed total domination in graphs. Journal of Beijing Institute of Technology 12 (2003), 319–321.
H. M. Xing, L. Sun and X. G. Chen: On a generalization of signed total dominating functions of graphs. Ars Combin. 77 (2005), 205–215.
B. Zelinka: Signed total domination number of a graph. Czech. Math. J. 51 (2001), 225–229.
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Xing, HM., Liu, HL. Minus total domination in graphs. Czech Math J 59, 861–870 (2009). https://doi.org/10.1007/s10587-009-0060-0
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DOI: https://doi.org/10.1007/s10587-009-0060-0