Connectivity in Matroids

Abstract

An edge of a 3-connected graph G is called essential if the 3-connection of G is destroyed both when the edge is deleted and when it is contracted to a single vertex. It is known (1) that the only 3-connected graphs in which every edge is essential are the “wheel-graphs.” A wheel-graph of order n, where n is an integer ⩾3, is constructed from an n -gon called its “rim” by adding one new vertex, called the “hub,” and n new edges, or “spokes” joining the new vertex to the n vertices of the rim; see Figure 4A. A matroid can be regarded as a generalized graph. One way of developing the theory of matroids is therefore to generalize known theorems about graphs. In the present paper we do this with the theorem stated above.