Upper bounds on the upper signed total domination number of graphs

Let G = (V, E) be a graph. A function f : V → {- 1, + 1} defined on the vertices of G is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. A signed total dominating function f is minimal if there does not exist a signed total dominating function g, f ≠ g, for which g (v) ≤ f (v) for every v ∈ V. The weight of a signed total dominating function is the sum of its function values over all vertices of G. The upper signed total domination number of G is the maximum weight of a minimal signed total dominating function on G. In this paper we present a sharp upper bound on the upper signed total domination number of an arbitrary graph. This result generalizes previous results for regular graphs and nearly regular graphs.