Total κ-subdominating functions on graphs

A two-valued function f defined on the vertices of a graph G = (V,E), f: V -→ {- 1,1}, is an opinion function. For each vertex v of G, the vote of v is the sum of the function values of / over the open neighborhood of v. A total fc-subdominating function (TkSF) of a graph G is an opinion function for which at least k of the vertices have a vote value of at least one. The total fc-subdomination number, γ^t_kS (G), of G is the minimum value of f(V) over all TkSF s of G where f(V) denotes the sum of the function values assigned to the vertices under /. We give a lower bound on _S(G) in terms of the minimum degree, maximum degree and the order of G. A lower bound on γ^t_kS (G) in terms of the degree sequence of G is given. Lower and upper bounds on γ^t_kS(G) for a tree G are presented.