The cost number and the determining number of a graph

Abstract

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 ρ d (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 obtain some general upper and lower bounds for ρ d (G) based on Det(G). Finally, we compute the cost and the determining number for the friendship graphs and corona product of two graphs.

Paint Cost and the Frugal Distinguishing Number

“…We can ask how close G is to being (d−1)distinguishable (which is the size of a smallest color class) or we can ask how many vertices we need to (re)color to achieve a d-distinguishing coloring (which is the size of the smallest complement of a color class). In [4], Alikhani and Soltani take the first approach by defining the cost number, denoted ρ d (G), to be the size of a smallest color class over all d-distinguishing colorings of G. In this paper we take the second approach by defining the paint cost of d-distinguishing G, or equivalently the d-paint cost, denoted ρ d (G), to be the minimum size of the complement of a color class over all d-distinguishing colorings. As one expects from generalizations, if…”