Stable sets versus independent sets

Let G=(V, E) be a graph and let L(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 L(G)=J1∪J2∪⋯∪Jm for some matroids Mi=(V,Ji)(i=1,2,...,m). We characterize the graphs of matroidal number at most m for all m≥1. For m≤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 \u27union\u27 by \u27intersection\u27. © 1993