 By Wai-Kai Chen

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Pictorially, the meaning is obvious. For example, in fig. 2 . 7 , the orientations of the edge e2 = (l92) and the cutsets 4 and 7 coincide, and the orientations of edge e2 and the cutset 1 are opposite. R LEMMA 2 . 4 : There are ( 2 — 1 ) nonempty cuts in G. 5 : A cut and a circuit of G have an even number of edges in common. The above two lemmas follow directly from our interpretation of a cut. 5). Once again, we can discuss cuts most conveniently by means of a cut matrix. 15: Cut-edge incidence matrix (cut matrix).

Let Gs be a subgraph of a connected graph G. Show that the subgraph Gs cannot have more components than there are nodes in Gs. 3. 1. 4. Let P' and P" be two different paths connected between the nodes / and j . Show that there is a circuit in P ' u P " . Is there a circuit containing the node / or j or b o t h ? 5. 3. 6. Show that an /2-node graph which has no parallel edges and self-loops and which has more than ^(n — 1) (n — 2) edges is always connected. 7. 3 is an equivalence relation. 8. 5.

Then the set of edges el9 e59 e39 and e6 forms a cut of G. On the other hand, if we let ^ = { 3 , 4 } and V2 = {\,2, 5 } , the set of edges el9 e5, and e3 forms a cutset. Of course, the cutsets and the incidence cuts are special types of cuts. Like the circuit, it is more natural to orient the cuts in a directed graph. DEFINITION 2 . 1 4 : Cut orientation. For a directed graph G, let V1 and V2 be the sets of nodes partitioned by a cut C of G. The cut C is said to be oriented if the sets Vt and V2 are ordered either as (Vl9 V2) or as (V29 V^).