Remarks on the minus (signed) total domination in graphs

A function f : V (G) → { + 1, 0,- 1 } defined on the vertices of a graph G is a minus total dominating function if the sum of its function values over any open neighborhood is at least 1. The minus total domination number γt-(G) of G is the minimum weight of a minus total dominating function on G. By simply changing "{ + 1, 0,- 1 }" in the above definition to "{ + 1,- 1 }", we can define the signed total dominating function and the signed total domination number γts(G) of G. In this paper we present a sharp lower bound on the signed total domination number for a k-partite graph, which results in a short proof of a result due to Kang et al. on the minus total domination number for a k-partite graph. We also give sharp lower bounds on γtsand γt-for triangle-free graphs and characterize the extremal graphs achieving these bounds.