Lower bounds on the minus domination and <f>k</f>-subdomination numbers

A three-valued function <f>f</f> defined on the vertex set of a graph <f>G=(V,E)</f>, <f>f : V→{−1,0,1}</f> is a minus dominating function if the sum of its function values over any closed neighborhood is at least one. That is, for every <f>v∈V</f>, <f>f(N[v])&ges;1</f>, where <f>N[v]</f> consists of <f>v</f> and all vertices adjacent to <f>v</f>. The weight of a minus function is <f>f(V)=∑v∈Vf(v)</f>. The minus domination number of a graph <f>G</f>, denoted by <f>γ−(G)</f>, equals the minimum weight of a minus dominating function of <f>G</f>. In this paper, sharp lower bounds on minus domination of a bipartite graph are given. Thus, we prove a conjecture proposed by Dunbar et al. (Discrete Math. 199 (1999) 35), and we give a lower bound on <f>γks(G)</f> of a graph <f>G</f>. [Copyright &y& Elsevier]