see this, let D be a dominating set in a claw-free graph, and suppose that v and w are two adjacent vertices in D; then the set of vertices dominated by v but not by w must be a clique (else v would be the center of a claw). If every vertex in this clique is already dominated by at least one other member of D, then v can be removed producing a smaller independent dominating set, and otherwise v can be replaced by one of the undominated vertices in its clique producing a dominating set with fewer adjacencies. By repeating this replacement process one eventually reaches a dominating set no larger than D, so in particular when the starting set D is a minimum dominating set this process forms an equally small independent dominating set. Despite this domination perfectness property, it is NP-hard to determine the size of the minimum dominating set in a claw-free graph. However, in contrast to the situation for more general classes of graphs, finding the minimum dominating set or the minimum connected dominating set in a claw-free graph is fixed-parameter tractable: it can be solved in time bounded by a polynomial in the size of the graph multiplied by an exponential function of the dominating set size. Structure overview a series of papers in which they prove a structure theory for claw-free graphs, analogous to the graph structure theorem for minor-closed graph families proven by Robertson and Seymour, and to the structure theory for perfect graphs that Chudnovsky, Seymour and their co-authors used to prove the strong perfect graph theorem. The theory is too complex to describe in detail here, but to give a flavor of it, it suffices to outline two of their results. First, for a special subclass of claw-free graphs which they call quasi-line graphs (equivalently, locally co-bipartite graphs), they state that every such graph has one of two forms: A fuzzy circular interval graph, a class of graphs represented geometrically by points and arcs on a circle, generalizing proper circular arc graphs. A graph constructed from a multigraph by replacing each edge by a fuzzy linear interval graph. This generalizes the construction of a line graph, in which every edge of the multigraph is replaced by a vertex. Fuzzy linear interval graphs are constructed in the same way as fuzzy circular interval graphs, but on a line rather than on