Let G = (V,E) be a graph and let k ∈ Z⁺. A total k-subdominating function is a function f: V → {-1,1} such that for at least k vertices v of G, the sum of the function values of f in the open neighborhood of v is positive. The total k-subdomination number of G is the minimum value of f(V) over all total k-subdominating functions f of G where f(V) denotes the sum of the function values assigned to the vertices under f. In this paper, we present a cubic time algorithm to compute the total k-subdomination number of a tree and also show that the associated decision problem is NP-complete for general graphs.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1296,
author = {Laura M. Harris and Johannes H. Hattingh and Michael A. Henning},
title = {Algorithmic aspects of total-subdomination in graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {26},
year = {2006},
pages = {5-18},
zbl = {1103.05060},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1296}
}
Laura M. Harris; Johannes H. Hattingh; Michael A. Henning. Algorithmic aspects of total-subdomination in graphs. Discussiones Mathematicae Graph Theory, Tome 26 (2006) pp. 5-18. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1296/
[000] [1] L.W. Beineke and M.A. Henning, Opinion function on trees, Discrete Math. 167-168 (1997) 127-139, doi: 10.1016/S0012-365X(96)00221-X.
[001] [2] I. Broere, J.H. Hattingh, M.A. Henning and A.A. McRae, Majority domination in graphs, Discrete Math. 138 (1995) 125-135, doi: 10.1016/0012-365X(94)00194-N. | Zbl 0820.05037
[002] [3] G. Chang, S.C. Liaw and H.G. Yeh, k-subdomination in graphs, Discrete Applied Math. 120 (2002) 55-60, doi: 10.1016/S0166-218X(01)00280-3. | Zbl 1008.05110
[003] [4] E.J. Cockayne and C. Mynhardt, On a generalisation of signed dominating functions of graphs, Ars Combin. 43 (1996) 235-245. | Zbl 0881.05060
[004] [5] J.E. Dunbar, S.T. Hedetniemi, M.A. Henning and P.J. Slater, Signed domination in graphs, Graph Theory, Combinatorics and Applications, John Wiley & Sons, Inc. 1 (1995) 311-322. | Zbl 0842.05051
[005] [6] O. Favaron, Signed domination in regular graphs, Discrete Math. 158 (1996) 287-293, doi: 10.1016/0012-365X(96)00026-X. | Zbl 0861.05033
[006] [7] Z. Furedi and D. Mubayi, Signed domination in regular graphs and set-systems, J. Combin. Theory (B) 76 (1999) 223-239, doi: 10.1006/jctb.1999.1905. | Zbl 0933.05117
[007] [8] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, New York, 1979). | Zbl 0411.68039
[008] [9] L. Harris and J.H. Hattingh, The algorithmic complexity of certain functional variations of total domination in graphs, Australasian J. Combin. 29 (2004) 143-156. | Zbl 1084.05049
[009] [10] L. Harris, J.H. Hattingh and M.A. Henning, Total k-subdominating functions on graphs, submitted for publication. | Zbl 1097.05031
[010] [11] J.H. Hattingh, M.A. Henning and P.J. Slater, The algorithmic complexity of signed domination in graphs, Australasian J. Combin. 12 (1995) 101-112. | Zbl 0835.68089
[011] [12] M.A. Henning, Domination in regular graphs, Ars Combin. 43 (1996) 263-271. | Zbl 0881.05101
[012] [13] M.A. Henning, Dominating functions in graphs, Domination in Graphs: Volume II (Marcel-Dekker, Inc., 1998) 31-62. | Zbl 0891.05044
[013] [14] M.A. Henning, Signed total domination in graphs, to appear in Discrete Math.
[014] [15] M.A. Henning and H. Hind, Strict open majority functions in Graphs, J. Graph Theory 28 (1998) 49-56, doi: 10.1002/(SICI)1097-0118(199805)28:1<49::AID-JGT6>3.0.CO;2-F | Zbl 0919.05032
[015] [16] M.A. Henning and P.J. Slater, Inequalities relating domination parameters in cubic graphs, Discrete Math. 158 (1996) 87-98, doi: 10.1016/0012-365X(96)00025-8. | Zbl 0858.05058
[016] [17] P. Lam and B. Wei, On the total domination number of graphs, submitted for publication.
[017] [18] J. Matousek, On the signed domination in graphs, Combinatoria 20 (2000) 103-108, doi: 10.1007/s004930070034. | Zbl 0949.05058
[018] [19] X. Yang, X. Huang and Q. Hou, On graphs with equal signed and majority domination number, manuscript (personal communication by X. Yang).
[019] [20] H. Yeh and G.J. Chang, Algorithmic aspects of majority domination, Taiwanese J. Math. 1 (1997) 343-350. | Zbl 0882.05114
[020] [21] B. Zelinka, Some remarks on domination in cubic graphs, Discrete Math. 158 (1996) 249-255, doi: 10.1016/0012-365X(94)00324-C. | Zbl 0861.05034
[021] [22] B. Zelinka, Signed total domination number of a graph, Czechoslovak Math. J. 51 (2001) 225-229, doi: 10.1023/A:1013782511179. | Zbl 0977.05096
[022] [23] Z. Zhang, B. Xu, Y. Li and L. Liu, A note on the lower bounds of signed domination number of a graph, Discrete Math. 195 (1999) 295-298, doi: 10.1016/S0012-365X(98)00189-7. | Zbl 0928.05052