Infinite subgraphs as matroid circuits

Abstract

A matroidal family C is defined to be a collection of graphs such that, for any given graph G, the subgraphs of G isomorphic to a graph in C satisfy the matroid circuit-axioms. Here matroidal families closed under homeomorphism are considered. A theorem of Simöes-Pereira shows that when only finite connected graphs are allowed as members of C , two matroids arise: the cycle matroid and bicircular matroid. Here this theorem is generalized in two directions: the graphs are allowed to be infinite, and they are allowed to be disconnected. In the first case four structures result and in the second case two infinite families of matroids are obtained. The main theorem concerns the structures resulting when both restrictions are relaxed simultaneously.