The algorithmic complexity of certain functional variations of total domination in graphs.
Laura M. Harris, Johannes H. Hattingh · 2004
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 s 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, weshow that the decision problem corresponding to determining whether a graph has a total minus dominating function of weight at most k maybeNPcomplete, even when restricted to bipartite or chordal graphs. Linear time algorithms for computing γ − t (T)andγ s t (T) for an arbitrary tree T are also presented.