Upper bounds on the upper signed total domination number of graphs ?
Let G=(V,E) G = ( V , E ) be a graph. A function f:V?{?1,+1} f : V ? { ? 1 , + 1 } defined on the vertices of G 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 f is minimal  if there does not exist a signed total dominating function g g , f?g f ? g , for which g(v)?f(v) g ( v ) ? f ( v ) for every v?V v ? V . The weight  of a signed total dominating function is the sum of its function values over all vertices of G G . The upper signed total domination number  of G G is the maximum weight of a minimal signed total dominating function on G 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.
upper bound upper signed total domination dominating function