Connectivity of Isotropic Systems

Isotropic systems are structures that unify some properties of Cregular graphs and autodual properties of binary matroids. Thus an isotropic system can be associated to any pair of dual binary matroids and also to any 4-regular graph. However, there are isotropic systems that cannot be obtained in this way. Isotropic systems are introduced in [2] and [3]. The connectivity of a matroid in Tutte’s sense [9] is autodual. Thus we define for each integer k > 0 the notion of a k-connected isotropic system in such a way that any isotropic system S derived from a pair of dual binary matroids is kconnected if and only if these matroids are k-connected. If S is derived from some 4-regular graph G, then if S is k-connected, G is cyclically 2kedge connected. The converse is true if and only if k I 3. This is similar to the relation between the connectivity of a graph and those of its cycle-matroid. There is another connection between isotropic systems and graph theory. To any isotropic system S is associated a set of simple graphs called its fundamental graphs. If S is associated to a pair of dual binary matroids ( M , M*), then we obtain a fundamental graph (but not all) of S by constructing a bipartite graph whose color classes are a base B and the complementary cobase B’, with an edge xy, x E B, y E B’, if and only if x belongs to the fundamental circuit of M contained in B + { y } . If F is a fundamental graph of an isotropic system S, then we show that S is 3-connected if and only if F is prime in the sense of Cunningham’s decomposition theory of graphs [S]. A theorem of Tutte [lo] says that if a matroid is 3-connected, then it has some 3-connected minor with one cell less, unless it is either a wheel or a whirl. We show that a similar result for 3-connected isotropic systems can be derived from a reduction theorem for prime graphs stated in [4]. But isotropic systems have three minors at each cell when matroids have two. This implies that we find a single minimal 3-connected isotropic system instead of the two infinite families for matroids. If the 3-connected system S derives from a cyclically 6-edge connected +regular graph G, the interpretation of the preceding theorem is obtained by introducing j-minors. A j-minor of G at a vertex u is obtained by splitting v into two vertices of degree 2 and contracting one edge incident to each of these vertices in order to get a new 4-regular graph. The interpretation is that unless G is a complete graph of order 5, then one of its j-minors is cyclically 6-edge connected. We give a direct proof of this result in the last section of this paper.