Signed total domination in nearly regular graphs

Abstract

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. An STDF f is minimal if there does not exist a STDF g:V(G)→{−1,l}, f ≠ g, for which g(v)≤f(v) for every v e V(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 domination number of G is the maximum weight of a minimal STDF of G. In this paper, we present sharp upper bounds on the upper signed total domination number of a nearly regular graph.