1. Vertex Cut
Vertex Cut: A separating set or vertex cut of a graph G is a set SV(G) such that S has more than one component.
Connectivity of G ((G)): The minimum size of a vertex set S such that S is disconnected or has only one vertex.
k-Connected Graph: The graph whose connectivity is at least k.

2. Connectivity of Kn and Km,n
A clique has no separating set. And, Kn- S has only one vertex for S=Kn-1  (Kn)=n-1.
Every induced subgraph that has at least one vertex from X and from Y is connected.  Every separating set contains X or Y
 (Km,n)= min(m,n) since X and Y themselves are separating sets (or leave only one vertex).

3. Connectivity of Qk
K-dimensional Hypercube Qk :

4. Connectivity of Qk (2/3)
For k>=2, the neighbors of one vertex in Qk form a separating set.  (Qk)<=k.
Every vertex cut has size at least k as proved by induction on k.  (Qk) =k.
Basic Step: For k<=1, Qk is a complete graph with k+1 vertices and has connectivity k.
Induction Hypothesis: (Qk-1)=k-1.
Induction Step: Consider as two copies Q and Q’ of Qk-1 plus a matching that joins corresponding vertices in Q and Q’. Let S be a vertex cut in Qk.

5. Connectivity of Qk (3/3)
Case 1: Q-S is connected and Q’-S is connected.  S contains at least one endpoint of every match pair.  |S|>=2k-1.  |S|>=k for k>=2.
Case 2: Q-S is disconnected.  S contains at least k-1 vertices in Q by induction hypothesis.  |S|>=k. Otherwise, S contains no vertices of Q’, implying Qk-S is connected.

6. Harary Graph Hk,n
Given 2<=k<n, place n vertices around a circle, equally spaced.
Case 1: k is even. Form Hk,n by making each vertex adjacent to the nearest k/2 vertices in each direction around the circle.
Case 2: k is odd and n is even. Form Hk,n by making each vertex adjacent to the nearest (k-1)/2 vertices in each direction around the circle and to the diametrically opposite vertex.

7. Harary Graph Hk,n (2/2)
Case 3: k is odd and n is odd. Index the vertices by the integers modulo n. Form Hk,n by making each vertex adjacent to the nearest (k-1)/2 vertices in each direction around the circle and adding the edges ii+(n-1)/2 for 0<=i<=(n-1)/2.

8. Theorem 4.1.5
(Hk,n ) =k, and hence the minimum number of edges in a k-connected graph on n vertices is kn/2.
Proof. 1. (Hk,n ) =k is proved only for the even case k=2r. (Leave the odd case as Exercise 12)
2. Consider SV(G) with |S|<k.
3. Consider u,vS. The original circular has a clockwise u,v-path and a counterclockwise u,v-path along the circle. Let A and B be the sets of internal vertices on these two paths.

9. Theorem (2/2)
4. |S|<k.  S has fewer than k/2 vertices in one of A and B, say A.  Deleting fewer than k/2 consecutive vertices cannot block travel in the direction of A.  There is a u,v-path in S via the set A.  S is connected.  S is not a vertex cut.
5. (Hk,n )>=k.  (Hk,n) =k since (G)=k.
6. Each vertex has k incident edge in k-connected graph.  k-connected graph on n vertices has at least kn/2 vertices.
7. Hk,n has kn/2 edges.  The minimum number of edges in a k-connected graph on n vertices is kn/2.

10. Edge Cut
Disconnecting Set of Edges: A set of edges F such that G-F has more than one component.
K-Edge-Connected Graph: Every disconnecting set has at least k edges.
Edge-Connectivity of G (’(G)): The minimum size of a disconnecting set.
Edge Cut: Given S,TV(G), [S,T] denotes the set of edges having one endpoint in S and the other in G. An edge cut is an edge set of the form [S, S], where S is a nonempty proper subset of V(G).

11. Edge Cut (2/2)

12. Theorem 4.1.9 If G is a simple graph, then (G)<=’(G)<= (G).
Proof. 1. The edges incident to a vertex v of minimum degree form an edge cut.  ’(G)<= (G).
2. Consider a smallest edge set [S, S].
3. If every vertex of S is adjacent to every vertex of S, then ’(G)=|[S, S]|=|S||S|>=n(G)-1.  ’(G)>=k(G) since (G)<=n(G)-1.
4. Otherwise, we choose xS and y S with (x,y)E(G).

13. Theorem 4.1.9
5. Let T consist of all neighbors of x in S and all vertices of S-{x} with neighbors in S.
6. Every x,y-path pass through T.  T is a separating set.  (G)<=|T|.
7. Pick the edges from x to T S and one edge from each vertex of TS to S yields |T| distinct edges of [S, S].  ’(G)= |[S, S]|>=|T|.  (G)<=’(G).

14. Possibility of (G)<’(G)<(G)

15. Theorem 4.1.11 If G is a 3-regular graph, then (G) =’(G).
Proof. 1. Let S be a minimum vertex cut.
2. Let H1, H2 be two components of G-S.
3. Each vS has a neighbor in H1 and a neighbor in H2. Otherwise, S-{v} is a minimum vertex cut.
4. G is 3-regular, v cannot have two neighbors in H1 and two in H2.
5. For each vS with one neighbor in S, delete the edge from v to a member of {H1, H2} where v has only one neighbor.

16. Theorem (2/2)
6. For each vS with no neighbor in S, delete the edge to H1.
7. These (G) edges break all paths from H1 to H2 .