William T. Tutte (1917–2002), Volume 51, Number 3

ing from the behavior of linearly independent sets of columns of a matrix, in 1935, Whitney defined a matroid M to consist of a finite set E (or E(M)) and a collection I (or I(M)) of subsets of E called independent sets with the properties that the empty set is independent; every subset of an independent set is independent; and if one independent set has more elements than another, then an element can be chosen from the larger set to adjoin to the smaller set to produce another independent set. A set that is not independent is called dependent, and minimal such sets are called circuits. For a matrix A over a field F, if E is the set of column labels on A and I is the set of subsets of E that label linearly independent sets of columns, then (E, I) is a matroid, M[A]. Such a matroid is said to be representable over the field F. If G is a graph, E is its set of edges, and I is the collection of the edge sets of the forests in G , then (E, I) is a matroid, M(G) , called the cycle matroid of G . Its circuits are the edge sets of the cycles inG . A matroid that is isomorphic to the matroid M(G) for some graph G is called graphic. Representable and graphic matroids were introduced by Whitney, and these classes have motivated much of the development of matroid theory ever since. A graph is planar if it can be drawn in the plane without edges crossing, and it is a plane graph if it is so drawn in the plane. The drawing separates the rest of the plane into regions called faces. Every plane graph G has a dual graph G∗ , formed by introducing a vertex of G∗ for each face of G and joining two vertices of G∗ by k edges if and only if the corresponding faces of G share k edges in their boundaries. Whitney defined the dual of a graph abstractly and showed that a graph has such aly and showed that a graph has such a dual if and only if it is a planar graph. For arbitrary matroids there is a natural notion of a dual matroid that extends the notion of duality for planar graphs. For a matroid M on E, let B be the collection of bases, that is, maximal independent sets, of M . Clearly B uniquely determines M. The collection {E − B : B ∈ B} is the set of bases of another matroid on E, namely, the dual M∗ of M . Evidently,