need some proof and we shall work o

need some proof and we shall work out the details in Chapter 7.) Since T does not contain any loops we deduce that r must be connected. In fact r is a tree. For if there were a loop in r it would separate P into two distinct pieces by hypothesis (b), and each of these pieces must contain at least one vertex of T. Any attempt to connect two vertices of T which lie in different pieces by a chain of edges results in a chain which meets this separating loop, and therefore in a chain which cannot lie entirely in T. This contradicts the fact that T is connected. Therefore r is a tree. (The proof breaks down here for a polyhedron such as that shown in Fig. 1.3, because the dual graph r will contain loops.)