Domination in regular graphs

A two-valued function f defined on the vertices of a graph G = (V, E), f: V --> {-1, 1}, is a signed dominating function if the sum of its function values over any closed neighborhoods is at least one. That is, for every v is an element of V, f(N[v]) greater than or equal to 1, where N[v] consists of v and every vertex adjacent to v. The function f is a majority dominating function if for at least half the vertices v is an element of V, f(N[v]) greater than or equal to 1. The weight of a signed (majority) dominating function is f(V) = Sigma f(v), over all vertices v is an element of V. The signed (majority) domination number of a graph G, denoted gamma(s)(G) (gamma(maj)(G), respectively), equals the minimum weight of a signed (majority, respectively) dominating function of G. In this paper, we establish an upper bound on gamma(s)(G) and a lower bound on gamma(maj)(G) for regular graphs G.