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A function f: V \rightarrow \{-1,0,1\} is a minus-domination function of agraph G=(V,E) if the values over the vertices in each closed neighborhood sumto a positive number. The weight of f is the sum of f(x) over all vertices x\in V. The minus-domination number \gamma^{-}(G) is the minimum weight over allminus-domination functions. The size of a minus domination is the number ofvertices that are assigned 1. In this paper we show that the minus-dominationproblem is fixed-parameter tractable for d-degenerate graphs when parameterizedby the size of the minus-dominating set and by d. The minus-domination problemis polynomial for graphs of bounded rankwidth and for strongly chordal graphs.It is NP-complete for splitgraphs. Unless P=NP there is no fixed-parameteralgorithm for minus-domination. 79,1 5%

On Complexities of Minus Domination