Geodesic
Algorithmic aspects of total-subdomination in graphs
Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 1, pp. 5-18

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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.
Keywords: total k-subdomination, algorithm, tree 
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Harris, Laura; Hattingh, Johannes; Henning, Michael. Algorithmic aspects of total-subdomination in graphs. Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 1, pp. 5-18. http://geodesic.mathdoc.fr/item/DMGT_2006_26_1_a0/