Symmetric Matroids and Connectivity Properties of Graphs

A symmetric matroid is a matroid deened on the edge-set of some countably innnite complete graph K in a way that ranks of nite sub-graphs of K are invariant under isomorphism. Thus a symmetric ma-troid M induces on any nite graph G a uniquely determined matroid M(G). We study connectivity properties of circuits and generalized trees of symmetric matroids. We give several characterizations of a special class of symmetric matroids called connectivity matroids. These matroids play for generalized notions of higher vertex-connectivity for graphs a role similar to the role the polygon-matroid plays for simple connectivity. Our theorem generalizes earlier results by G. Kalai. Using our methods we also give a simple proof of a well-known theorem by Simoes-Pereira on matroidal families.