Some relations between the distinguishing and some graph parameters

The distinguishing number of a simple graph $G$ is the least number $D(G)$ of colors needed for a coloring of $G$ which is preserved only by the identity automorphism. ##Similar parameters have been defined whose concern is breaking the symmetries of a graph. ##In this paper, we present interesting connections between these parameters and some other graph parameters such as the independence number. In particular, we study conditions under which a given graph $G$ is $(D,alpha)$-ordinary, that is, for which $D(G)leq alpha(G)$.