THE COST NUMBER an‎d THE DETERMINING NUMBER OF A GRAPH

The distinguishing number D(G) of a graph G is the least integer d such that G has an vertex labeling with d labels that is preserved only by a trivial automorphism. The minimum size of a label class in such a labeling of G with D(G)=d is called the cost of d-distinguishing G and is denoted by ρd(G). A set of vertices S⊆V(G) is a determining set for G if every automorphism of G is uniquely determined by its action on S. The determining number of G, Det(G), is the minimum cardinality of determining sets of G. In this paper we compute the cost and the determining number for the friendship graphs and corona product of two graphs.