Remarks on the minus (signed) total domination in graphs

A function image 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 image of G is the minimum weight of a minus total dominating function on G. By simply changing “image” in the above definition to “image”, we can define the signed total dominating function and the signed total domination number image 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 image and image for triangle-free graphs and characterize the extremal graphs achieving these bounds.