Domination good vertices in graphs

Abstract

A vertex that is contained in some minimum dominating set of a graph G is a good vertex, otherwise it is bad. Let g(G) (respectively, b(G)) denote the number of good (respectively, bad) vertices in a graph G. We determine for which triples (x, y, z) there exists a graph G such that γ(G) = x, g(G) = y, and b(G) = z. Then we give graphs realizing these triples. Also, we show that no graph has g(G) = b(G) = γ(G) and characterize the graphs G for which g(G) = b(G) = γ(G) + 1.