Matroids and graphs

Abstract

the circuits of any finite graph G define a matroid. We call this the circuitmatroid and its dual the bond-matroid of G. In the present paper we determine a necessary and sufficient condition, in terms of matroid structure, for a given matroid M to be graphic (cographic), that is the bond-matroid (circuitmatroid) of some finite graph. The condition is that M shall be regular and shall not contain, in a sense to be explained, the circuit-matroid (bondmatroid) of a Kuratowski graph, that is a graph with one of the structures shown in Figure I. Some of the intermediate results seem to be of interest in themselves. These include the theory of dual matroids in ?2, Theorem (7.3) on regular matroids, and Theorem (8.4) on the bond-matroids of graphs. Our main theorem is evidently closely related to the theorem of Kuratowski on planar graphs [1]. This states that a graph is planar (i.e., can be imbedded in the plane) if and only if it contains no graph with the point set structure of a Kuratowski graph. Indeed it is not difficult to prove Kuratowski's Theorem from ours, using the principle that a graph is planar if and only if it has a dual graph, that is if and only if its circuit-matroid is graphic. However this paper is long enough already and we refrain from adding matter not essential to the proof and understanding of the main theorem. 2. Dual matroids. We define a matroid M on a set M, its flats and their dimensions as in HI, ?1. We call the dimension of the largest flat (M) also the dimension dM of the matroid. (See HI, ?2, for the notation (S)). We write ae(S) for the number of elements of any finite set S. We proceed to give a definition of the dual of M analogous to the definition of a dual vector space in terms of orthogonality. First, in analogy with A, ?4, we define a dendroid of M as a minimal sub