1. EMIS 8374 Vertex Connectivity Updated 20 March 2008

2. Slide 2 Network Connectivity An s,t vertex separator of a graph G = (V, E) is a set of vertices whose removal disconnects vertices s and t. The s,t-connectivity of a graph G is the minimum size of an s,t vertex separator. The vertex-connectivity of G is the min{s,t-connectivity of G: (s, t) in V}.

3. Slide 3 Example Graph 1 2 3 4 5 6 7 8

4. Slide 4 2-7 Vertex Separator {3, 4, 6} 1 2 3 4 5 6 7 8

5. Slide 5 2-7 Vertex Separator {1, 4, 8} 1 2 3 4 5 6 7 8 2,7-connectivity = 3

6. Slide 6 1-8 Vertex Separator {4,5,6} 1 2 3 4 5 6 7 8

7. Slide 7 1-8 Vertex Separator {2,3} 1,8-connectivity = 2 Vertex-Connectivity = 2 1 2 3 4 5 6 7 8

8. Slide 8 Menger’s Theorem Given and undirected graph G and two nonadjacent vertices s and t, the maximum number of vertex-disjoint (aside from sharing s and t) paths from s to t is equal to the s,t-connectivity of G.

9. Slide 9 Maximum Flow Formulation For s-t connectivity For G = (V, E), construct the network G' = (N, A) as follows –For each vertex v in V/{s, t}, add nodes v and v' –Add arc (v, v') with capacity 1 –For each edge (u, v) in E, add arcs (u', v) and (v', u) with infinite capacity –For each edge (s, v) in E, add arc (s, v) with infinite capacity –For each edge (v, t) in E, add arc (v', t) with infinite capacity

10. Slide 10 Network for Example Graph 1 8 22'2'33'3'44'4'55'5'66'6'77'7' 11 1 1 1 1

11. Slide 11 Maximum 1-8 Flow 1 8 22'2' 33'3' 44'4'55'5' 66'6' 77'7' 11 1 1 1 1

12. Slide 12 Proof of Menger’s Theorem Lemma 1 –Each set of k vertex-disjoint s,t paths in G, corresponds to exactly one integral flow of value k in G'. Lemma 2 –Each s,t cut of finite capacity c corresponds to an s,t vertex-separator of size c in G Result follows from Max-Flow Min-Cut Theorem

13. Slide 13 Finding the Vertex-Connectivity of G = (N, A) Let Node 1 be the source node s Let c = |N| For i = 2 … |N| –Let t = i –If s-t Connectivity < c Then Let c = s-t Connectivity Return c Find Vertex-Connectivity of G with |N|-1 Maximum Flow Computations