STABLE SETS VERSUS INDEPENDENT SETS

Let G = (V, E) be a graph and let J(G) be the set of stable sets of G. The matroidal number of G, denoted by m(G), is the smallest integer m such that J(G) = J1 or J2 or ... or J(m) for some matroids M(i) = (V, J(i))(i = 1, 2, ..., m). We characterize the graphs of matroidal number at most m for all m greater-than-or-equal-to 1. For m less-than-or-equal-to 3, we show that the graphs of matroidal number at most m can be characterized by excluding finitely many induced subgraphs. We also consider a similar problem which replaces 'union' by 'intersection'.