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The algorithmic complexity of certain functional variations of total domination in graphs

A two-valued function f defined on the vertices of a graph G = (V, E), f : V → {−1, 1}, is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. That is, for every v ∈ V, f(N(v)) ≥ 1, where N(v) consists of every vertex adjacent to v. The weight of a total signed dominating function is f(V ) = ∑ f(v), over all vertices v ∈ V . The total signed domination number of a graph G, denoted γ t (G), equals the minimum weight of a total signed dominating function of G. If, instead of the range {−1, 1}, we allow the range {−1, 0, 1}, then we get the concept of a total minus dominating function. Its associated parameter, called the total minus domination number of a graph G, is denoted γ− t (G). In this paper, we show that the decision problem corresponding to the computation of the total minus domination number of a graph is NP-complete, even when restricted to bipartite graphs or chordal graphs. For a fixed k, we show that the decision problem corresponding to determining whether a graph has a total minus dominating function of weight at most k may be NPcomplete, even when restricted to bipartite or chordal graphs. Linear time algorithms for computing γ− t (T ) and γ s t (T ) for an arbitrary tree T are also presented. 144 LAURA HARRIS AND JOHANNES H. HATTINGH