Abstract
A function f: V → { − 1,0,1} is a minus-domination function of a graph G = (V,E) if the values over the vertices in each closed neighborhood sum to a positive number. The weight of f is the sum of f(x) over all vertices x ∈ V. In the minus-domination problem, one tries to minimize the weight of a minus-domination function. In this paper, we show that (1) the minus-domination problem is fixed-parameter tractable for d-degenerate graphs when parameterized by the size of the minus-dominating set and by d, where the size of a minus domination is the number of vertices that are assigned 1, (2) the minus-domination problem is polynomial for graphs of bounded rankwidth and for strongly chordal graphs, (3) it is NP-complete for splitgraphs, and (4) unless P = NP there is no fixed-parameter algorithm for minus-domination.
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Faria, L., Hon, WK., Kloks, T., Liu, HH., Wang, TM., Wang, YL. (2013). On Complexities of Minus Domination. In: Widmayer, P., Xu, Y., Zhu, B. (eds) Combinatorial Optimization and Applications. COCOA 2013. Lecture Notes in Computer Science, vol 8287. Springer, Cham. https://doi.org/10.1007/978-3-319-03780-6_16
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DOI: https://doi.org/10.1007/978-3-319-03780-6_16
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