Continuum theory and graph theory: disconnection numbers

Let A ' be a non-degenerate compact connected metric space. It is proved that if there is an n, 2 < n < Ko, such that X — A is not connected for all A < = X of cardinality n, then X is a graph (that is, a one-dimensional compact connected polyhedron), and conversely. Furthermore, a formula involving only the edges and nodes of the graph X is found which calculates the smallest such n (which works for all A c X of cardinality «). As one of the consequences, it is shown that these are exactly five compact connected metric spaces for which the smallest such n is 3. A number of other results, as well as some new proofs of known theorems, are obtained. 1