the minus k-domination numbers in graphs

For any integer k≥1, a minus k-dominating function is a function f:v→{−1,0,1} satisfying ∑w∈n[v]f(w)≥k for every v∈v(g), where n(v)={u∈v(g)∣uv∈e(g)} and n[ v]=n(v)∪{v}. the minimum of the values of ∑v∈v(g)f (v), taken over all minus k-dominating functions f, is called the minus k-domination number and is denoted by γ−k(g). in this paper, we introduce the study of minus k-domination in graphs and present several sharp lower bounds on the minus k-domination number for general graphs.