Total minus domination in k-partite graphs

A function f   defined on the vertices of a graph G=(V,E),f:V→{-1,0,1}G=(V,E),f:V→{-1,0,1} is a total minus dominating function (TMDF) if the sum of its values over any open neighborhood is at least one. The weight of a TMDF is the sum of its function values over all vertices. The total minus domination number, denoted by γt-(G), of G is the minimum weight of a TMDF on G  . In this paper, a sharp lower bound on γt- of k-partite graphs is given.