Upper bounds on the upper signed total domination number of graphs

Abstract

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.