Vertex Cut Vertex Cut: A separating set or vertex cut of a graph G is a set SV(G) such that G-S has more than one component. d f b e a g c i h

Connectivity Connectivity of G ((G)): The minimum size of a vertex set S such that G-S is disconnected or has only one vertex. Thus, (G) is the minimum size of vertex cut. (X) (G)=4

k-Connected Graph k-Connected Graph: The graph whose connectivity is at least k. (G)=2 a b c d e f g h i G is a 2-connected graph Is G a 1-connected graph ?

Connectivity of Kn A clique has no separating set. And, Kn- S has only one vertex for S=Kn-1  (Kn)=n-1.

Connectivity of Km,n 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). K4,3

Connectivity of Qk K-dimensional Hypercube Qk : K=1 K=2 K=3 K=4

Connectivity of Qk For k>=2, the neighbors of one vertex in Qk form a separating set.  (Qk)<=k. K=1 K=2 K=3 K=4

Connectivity of Qk 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.

Connectivity of Qk 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. Q Q’

Connectivity of Qk Case 2: Q-S is disconnected .  S contains at least k-1 vertices in Q. (Induction Hypothesis)  |S|>=k because S contains at least 1 vertices in Q’. (If S contains no vertices of Q’, Qk-S is connected.) Q Q’

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. (Hk,n)=k. |E(Hk,n)|= kn/2 H4,8

Harary Graph Hk,n 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. (Hk,n)=k. |E(Hk,n)|= kn/2 H5,8

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. In all cases, (Hk,n)=k. 2 1 3 4 5 6 7 8 |E(Hk,n)|= kn/2 H5,9 (Hk,n)=k. |E(Hk,n)|= (kn+1)/2

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. We need to show SV(G) with |S|<k is not a vertex cut since (Hk,n)=k. H4,8

Theorem 4.1.5 3. Consider u,vV-S. The original circular has a clockwise u,v-path and a counterclockwise u,v-path along the circle. 4. Let A and B be the sets of internal vertices on these two paths. 5. It suffices to show there is a u,v-path in V-S via the set A or the set B if |S|<k. . H4,8 u v A B

Theorem 4.1.5 6. |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 V-S via the set A. u v A B H4,8

Theorem 4.1.5 7. Since Hk,n has kn/2 edges, we need to show a k-connected graph on n vertices has at least kn/2 edges. 8. Each vertex has k incident edge in k-connected graph.  k-connected graph on n vertices has at least kn/2 vertices.

Disconnecting Set Disconnecting Set of Edges: A set of edges F such that G-F has more than one component. Edge-Connectivity of G (’(G)): The minimum size of a disconnecting set. k-Edge-Connected Graph: Every disconnecting set has at least k edges.

Edge Cut 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,V-S], where S is a nonempty proper subset of V(G). S V-S

Theorem 4.1.9 If G is a simple graph, then (G)<=’(G)<= (G). Proof. 1. ’(G)<= (G) since the edges incident to a vertex v of minimum degree form an edge cut. 2. We need to show (G)<=’(G). 3. Consider a smallest edge cut [S,V-S]. (’(G)= |[S,V-S]|) 4. Case 1: Every vertex of S is adjacent to every vertex of V-S.  ’(G)=|[S,V-S]|=|S||V-S|>=n(G)-1.  ’(G)>=k(G) since (G)<=n(G)-1. 5. Case 2: there exists xS and yV-S such that (x,y)E(G).

Theorem 4.1.9 5. Case 2: there exists xS and yV-S such that (x,y)E(G). 6. Let T consist of all neighbors of x in V-S and all vertices of S-{x} with neighbors in V-S. 7. Every x,y-path pass through T.  T is a separating set.  (G)<=|T|. 8. It suffices to show |[S,V-S]|>=|T|. x T T V-S S T T y T

Theorem 4.1.9 9. Pick the edges from x to TV-S and one edge from each vertex of TS to V-S yields |T| distinct edges of [S,V-S]. 9. Pick the edges from x to TV-S and one edge from each vertex of TS to V-S yields |T| distinct edges of [S,V-S].  ’(G)= |[S,V-S]|>=|T|. x T T V-S S T T y T

Possibility of (G)<’(G)<(G)

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. S H1 H2

Theorem 4.1.11 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. There are three cases for v. v v v H1 H2 H1 H1 H1 H2 u Case 1 Case 2 Case 3

Theorem 4.1.11 (2/2) 5. For Cases 1 and 2, delete the edge from v to a member of {H1, H2} where v has only one neighbor. 6. For Case 3, delete the edge from v to H1 and the edge from v to H2. v v v H1 H2 H1 H1 H1 H2 u Case 1 Case 2 Case 3 7. These (G) edges break all paths from H1 to H2 .