A Graphical Representation of Matroids

A base graph of a matroid is the graph whose points are the bases of the matroid. Two bases are adjacent if they differ by exactly one element. A definition of equivalence of matroids is given and it is shown that two matroids are equivalent if and only if their base graphs are isomorphic. In particular, if M and M1 are nonseparable matroids with isomorphic base graphs, then M is isomorphic to either M1 or its dual. Thus, the study of matroids is reduced to the study of a class of graphs: the base graphs. A detailed investigation of the structure of neighborhoods in the base graph is carried out and this is used to establish the above result. Included in the result is a graphical classification of separable matroids which gives a new proof of Whitney's decomposition theorem.