Generalized domination and efficient domination in graphs Original Research Article

This paper generalizes dominating and efficient dominating sets of a graph. Let G be a graph with vertex set V (G). If ƒ: V (G) → Y, where Y is a subset of the reals, the weight of ƒ is the sum of ƒ(v) over all v ϵ V(G). If the closed neighborhood sum of ƒ(v) at every vertex is at least 1, then ƒ is called a Y-dominating function of G. If the closed neighborhood sum is exactly 1 at every vertex, then ƒ is called an efficient dominating function. Two Y-dominating functions are equivalent if they have the same closed neighborhood sum at every vertex of G. It is shown that if the closed neighborhood matrix of G is invertiable then G has an efficient Y-dominating function for some Y. It is also shown that G has an efficient Y-dominating function if and only if all equivalent Y-dominating functions have the same weight. Related theoretical and computational questions are considered in the special cases where Y = {−1, 1} or Y = {−1, 0, 1}.