Signed total domination in graphs

A function f:V(G)→{-1,1} defined on the vertices of a graph G is a signed total dominating function (STDF) if the sum of its function values over any open neighborhood is at least one. A STDF f is minimal if there does not exist a STDF g:V(G)→{-1,1}, f≠g, for which g(v)≤f(v) for every vV(G). The weight of a STDF is the sum of its function values over all vertices. The signed total domination number of G is the minimum weight of a STDF of G, while the upper signed total domination number of G is the maximum weight of a minimal STDF on G. In this paper we study these two parameters. In particular, we present lower bounds on the signed total domination number and upper bounds on the upper signed total domination number of a graph.