Remarks on the minus (signed) total domination in graphs

Abstract

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 γ t - ( 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 γ t s ( 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 γ t s and γ t - for triangle-free graphs and characterize the extremal graphs achieving these bounds.