which a vertex v in dominating set D is chosen and replaced with a neighbor u (u is not in D) such that the modified D is also a dominating set and this process can be repeated over any infinite sequence of choices of vertices v. Dominating and independent sets Dominating sets are closely related to independent sets: an independent set is also a dominating set if and only if it is a maximal independent set, so any maximal independent set in a graph is necessarily also a minimal dominating set. Domination by independent sets A dominating set may or may not be an independent set. For example, figures (a) and (b) above show independent dominating sets, while figure (c) illustrates a dominating set that is not an independent set. The independent domination number of a graph is the size of the smallest dominating set that is an independent set. Equivalently, it is the size of the smallest maximal independent set. The minimum in is taken over less elements (only the independent sets are considered), so for all graphs . The inequality can be strict - there are graphs for which . For example, let be the double star graph'' consisting of vertices , where . The edges of are defined as follows: each is adjacent to , is adjacent to , and is adjacent to each . Then since is a smallest dominating set. If , then since is a smallest dominating set that is also independent (it is a smallest maximal independent set). There are graph families in which , that is, every minimum maximal independent set is a minimum dominating set. For example, if is a claw-free graph. A graph is called a domination-perfect graph if in every induced subgraph of . Since an induced subgraph of a claw-free graph is claw-free, it follows that every claw-free graphs is also domination-perfect. For any graph , its line graph is claw-free, and hence a minimum maximal independent set in is also a minimum dominating set in . An independent set in corresponds to a matching in , and a dominating set in corresponds to an edge dominating set in . Therefore a minimum maximal matching has the same size as a minimum edge dominating set. Domination of independent sets The independence domination number of a graph is the maximum, over all independent sets of , of the smallest