Matroidal families were introduced by SimiSes-Ferefra [S]. Altb~ough we know uncountably many matroidai families of simple graphs and infinitely many matroidal families with multigraphs as members, it is an open question how one can find ail matroidal families. In this paper we give a solution of this problem by cxtain submodular fuoctiorls on the set of all finite graphs characterizing the families of connected graphs being matroidal families. 1. IntrQduction Let 9 be the set of all finite graphs, where loops and multiple edges are allowed, and &, c $ the set of all fluite simple graphs. For a graph G = (V, E) and for Xc E we denote the subgraph of G induced by the edges of X by G ] X. Further we denote the number of vertices by a(G), the number of edges by K(G) and the number of components by a(G). Deenition Ll. Let 111 be a finite set and % c 2M a system d non-empty subsets of A4 The pair (M, %) is called a matroid on M, if the following axioms hold: (cl) No element of '& contains properly another element of %. (C2) If C, C' E Q. C# C' and x E C n C', then (C U C') -(x} contains an element C" of CQ. The elements of % are called the circuits of the matroid (M, Ce). In this paper we presuppose a knowledge of matroid theory; our standard reference is Welsh [9:. D&B&~ 1.2 A matmidaI family of graphs is a non-empty set of finite connected graphs 9 such that, given any graph G, the edge-sets of the subgraphs of G isomorphic to members of 8 may be regarded as the circuits of a matroid on the edge-set of G, denoted by Jc1(G, 9) and called the 3-matroid of G. Analogously we may define matroidal families of simple graphs. Sim6es-Pereira [S] introduced mat&da1 families in 1972 and discovered four of them: 90, PI, OOlZ-365XB4/$3.00 @