1. Internally Disjoint Paths Internally Disjoint Paths : Two paths u to v are internally disjoint if they have no common internal vertex. u u v v Common internal vertex Internally disjoint paths

2. Theorem 4.2.2 A graph G having at least three vertices is 2- connected if and only if for each pair u,v  V(G) there exists internally disjoint u,v-paths in G. (Proof of if part) 1.It suffices to show for each pair u,v  V(G), deletion of any vertex in V(G) cannot separate u from v. 2. This is clearly true because G has internally disjoint u,v-paths.

3. Theorem 4.2.2 (Proof of only if part) 1.G is 2-connected. (Premise) 2.That G has internally disjoint u,v-paths is proved by induction on d(u,v). 3. Basis Step: d(u,v)=1. 3.1. The graph G-uv is connected since  ’(G)>=  (G)>=2. 3.2. There exists a u,v-path in G-uv, which is internally disjoint in G from the u,v-path formed by the edge uv itself.

4. Theorem 4.2.2 4. Induction Step: d(u,v)>1. 4.1. Let k=d(u,v). 4.2. Let w be the vertex before v on a shortest u,v-path.  d(u,w)=k-1. 4.3 G has internally disjoint u,w-paths P and Q. (Induction Hypothesis)

5. Theorem 4.2.2 vu w P Q 4.4. If v  V(P)  V(Q), then we find the desired paths in the cycle P  Q.

6. Theorem 4.2.2 4.5. Otherwise, G-w is connected 4.6. G-w contains a u,v-path R. 4.7. If R avoids P or Q, we are done. v u w Q P R since G is 2-connected.

7. Theorem 4.2.2 4.8. Otherwise, let z be the last vertex of R (before v) belonging to P  Q. We assume that z  P by symmetry. 4.9. We combine the u,z-subpath of P with the z,v-subpath of R to obtain a u,v-path internally disjoint from Q  wv. vu w z P Q R

8. Lemma 4.2.3 (Expansion Lemma) If G is a k-connected graph, and G’ is obtained from G by adding a new vertex y with at least k neighbors in G, then G’ is k-connected.

9. Theorem 4.2.4 For a graph G with at least three vertices, the following conditions are equivalent (and characterize 2- connected graphs). A)G is connected and has no cut-vertex. B)For all x,y  V(G), there are internally disjoint x,y- paths. C)For all x,y  V(G), there is a cycle through x and y. D)  (G)>=1, and every pair of edges in G lies on a common cycle.

10. Theorem 4.2.4 Proof. 1. Theorem 4.2.2 proves A  B. 2. For B  C, the cycles containing x and y corresponds to pairs of internally disjoint x,y-paths. 3. For D  C,  (G)>=1 implies that vertices x and y are not isolated.

11. Theorem 4.2.4 4. Consider edges incident to x and y. 5. Case 1: there are at least two such edges e and f. Then e and f lies on a common cycle.  There is a cycle through x and y. 6. Case 2: only one such edge e. Let f be an edge incident to the third vertex.  e and f lies on a common cycle.  There is a cycle through x and y. xy e f u x y z f y x e

12. Theorem 4.2.4 7. For C  D. G satisfies condition C.  G satisfies condition A.  G is connected.  (G)>=1. 8. We need to show any two edges, uv and xy, lie on a common cycle. 9. Add to G the vertices w with neighborhood {u,v} and z with neighborhood {x,y} to form G’.

13. Theorem 4.2.4 10. Since G is 2-connected, Lemma 4.2.3 implies G’ is 2-connected. 11. w and z lie on a cycle C in G’. 12. Since w,z each have degree 2, C must contain the paths u,w,v and x,z,y but not the edges uv or xy. 13. Replacing the path u,w,v and x,z,y in C with the edges uv and xy yields the desired cycle through uv and xy in G. u v x y w z

14. x,y-cut x,y-cut: Given x,y  V(G), a set S  V(G)-{x,y} is an x,y-separator or x,y-cut if G-S has no x,y-path.  (x,y): the minimum size of x,y-cut. (x,y): the maximum size of a set of pairwise internally disjoint x,y-paths.

15. Example 4.2.16 {b,c,z,d} is an x,y-cut of size 4.   (x,y)<=4. G has four internally disjoint x,y-paths.  (x,y)>=4. {b,c,x} is an w,z-cut of size 3.   (w,z)<=3. G has three internally disjoint w,z-paths.  (w,z)>=3.

16. Theorem 4.2.17 (Menger Theorem) If x,y are vertices of a graph G and xy  E(G), then  (x,y) = (x,y). Proof. 1. An x,y-cut must contain an internal vertex of every internally disjoint x,y-paths, and no vertex can cut two internally disjoint x,y-paths.   (x,y)>= (x,y). 2. We prove equality by induction on n(G).

17. Theorem 4.2.17 (Menger Theorem) Basis Step: n(G)=2. xy  E(G) yields  (x,y)= (x,y)=0. Induction Step: n(G)>2. 1.Let k=  G (x,y). 2.No minimum cut properly contains N(x) or N(y) since N(x) and N(y) are x,y-cuts. 3. Case 1: G has a minimum x,y-cut S other than N(x) or N(y). 4. Case 2: Every minimum x,y-cut is N(x) or N(y).

18. Theorem 4.2.17 5. For Case 1, let V1 be the set of vertices on x,S-path, and let V2 be the set of vertices on S,y-path. 6. S  V1 and S  V2  S  V1  V2. 7. If there exists v such that v  V1  V2 –S, then combing x,v-portion of some x,S-path and v,y-portion of some S,y-path yields an x,y-path that avoids the x,y-cut S. It contradicts that S is a minimum x,y-cut. 8. This implies S=V1  V2. G V1V1 V2V2 xy S v

19. Theorem 4.2.17 H1 y’ x H2 x’ y 9. Form H1 by adding to G[V1] a vertex y’ with edges from S, and form H2 by adding to G[V2] a vertex x’ with edges from S.

20. Theorem 4.2.17 10. Every x,y-path in G starts with an x,S-path (contained in H1).  Every x,y’cut in H1 is an x,y-cut in G.   H1 (x,y’)= k. 11.  H2 (x’,y)= k by the same argument in 10. 12. H1 and H2 are smaller than G since N(y)  S and N(x)  S.  H1 (x,y’)=k= H2 (x’,y).

21. Theorem 4.2.17 13. S=V1  V2.  Deleting y’ from the k paths in H1 and x’ from the k paths in H2 yields the desired x,S-paths and S,y-paths in G that combine to form k internally disjoint x,y- paths in G. 

22. Theorem 4.2.17 14. For Case 2, if there exists node u  N(x)  N(y), then S-u is x,y-cut in G-u.   G-u (x,y)=k-1.  G-u has k-1 internally disjoint x,y-paths by induction hypothesis. 15. Combining these k-1 x,y-path and the path x,u,y yields k internally disjoint x,y-paths in G.

23. Theorem 4.2.17 16. If there exists node v  {x}  N(x)  N(y)  {y}, then S is minimum x,y-cut in G-v. (If there exists a x,y-cut, S’, in G-v whose size is smaller than |S|, then S’  {v} is a x,y-cut in G. It is a contradiction.)   G-v (x,y)=k.  G-v has k internally disjoint x,y-paths by induction hypothesis.  These are k internally disjoint x,y-paths in G.

24. Theorem 4.2.17 17. We may assume that N(x) and N(y) partition V(G)- {x,y}. 18. Let G’ be the bipartite graph with bipartition N(x), N(y) and edge set [N(x),N(y)]. 19. Every x,y-path in G uses some edge from N(x) to N(y).  x,y-cuts in G are the vertex covers of G’.   (G’)=k.  G’ has a matching of size k by Theorem 3.1.16.  These k edges yield k internally disjoint x,y-paths of length 3.

25. Line Graph (Digraph) Line Graph (Digraph): The line graph (digraph) of a graph (digraph) G, written L(G), is the graph (digraph) whose vertices are the edges of G, with ef  E(L(G)) when e=uv and f=vw in G.

26. Theorem 4.2.19 If x and y are distinct vertices of a graph or digraph G, then the minimum size of an x,y-disconnecting set of edges equals the maximum number of pairwise edge- disjoint x,y-paths. x y a b c d e f g ts Proof. 1. Modify G to obtain G’ by adding two new vertices s, t and two new edges sx and yt. 2. Cleary,  ’ G (x,y)=  ’ G’ (x,y) and ’ G (x,y)= ’ G’ (x,y).

27. Theorem 4.2.19 3. A x,y-path exists in G’ that traverses edges p, q, r if and only if a sx,yt-path exists in L(G’) that traverses vertices p, q, r.

28. Theorem 4.2.19 4. Edge-disjoint x,y-paths in G’ become internally disjoint sx,yt-paths in L(G’), and vice versa.  ’ G’ (x,y)= L(G’) (sx,yt). 5. A set of edges disconnects y from x in G’ if and only if the corresponding vertices of L(G’) form an sx,yt-cut.   ’ G’ (x,y)=  L(G’) (sx,yt). 6.  L(G’) (sx,yt)= L(G’) (sx,yt).   ’ G’ (x,y)= ’ G’ (x,y).   ’ G (x,y)= ’ G (x,y).

29. Lemma 4.2.20 Deletion of an edge reduces connectivity at most 1. Proof. 1.  (G-xy) ≤  (G). 2.  (G-xy)<  (G) only if G-xy has a separating set S that has size less than  (G) and is not a separating set of G. 3. G-S-xy has two components G[X] and G[Y], with x  X and y  Y since G-S is connected. 4. If |X| ≥ 2 (or |Y| ≥ 2), S  {x} (or S  {y}) is a separating set of G, and  (G) ≤  (G-xy)+1. 5. Otherwise, |S|=n(G)-2. 6. Since |S|<  (G),  (G) = n(G)-1. 7. G is complete graph and  (G-xy) = n(G)-2 =  (G)-1.

30. Theorem 4.2.21 The edge-connectivity of G equals the maximum k such that ’(x,y)>=k for all x,y  V(G). That is,  ’(G)= min x,y  V(G) ’(x,y). Proof. 1.  ’(G)= min x,y  V(G)  ’(x,y). 2.  ’(x,y)= ’(x,y) for all x,y  V(G) by Theorem 4.2.19.

31. Theorem 4.2.21 The connectivity of G equals the maximum k such that (x,y)>=k for all x,y  V(G). That is,  (G)=min x,y  V(G) (x,y). Proof. 1.  (G)=min xy  E(G)  (x,y). 2.  (x,y)= (x,y) for xy  E(G). 3. It suffices to show (x,y)≥  (G) if xy  E(G). 4. G-xy (x,y)=  G-xy (x,y) ≥  (G-xy) by Menger’s Theorem. 5.  (G-xy) ≥  (G) -1 by Lemma 4.2.20. 6. (x,y)= G-xy (x,y)+1 ≥  (G).