Graphs with equal domination and certified domination numbers

A set D of vertices of a graph G = (VG, EG) is a dominating set of G if every vertex in VG — D is adjacent to at least one vertex in D. The domination number (upper domination number, respectively) of G, denoted by [formula], respectively), is the cardinality of a smallest (largest minimal, respectively) dominating set of G. A subset D ⊆ VG is called a certified dominating set of G if D is a dominating set of G and every vertex in D has either zero or at least two neighbors in VG — D. The cardinality of a smallest (largest minimal, respectively) certified dominating set of G is called the certified (upper certified, respectively) domination number of G and is denoted by [formula]), respectively). In this paper relations between domination, upper domination, certified domination and upper certified domination numbers of a graph are studied.