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

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. () Consider any two vertices u,vV(G).  G has internally disjoint u,v-paths.  Deletion of any vertex in V(G) cannot separate u from v.  G is 2-connected.
() Suppose G is 2-connected. That G has internally disjoint u,v-paths is proved by induction on d(u,v).

3. Theorem 4.2.2 Basis Step: d(u,v)=1. since ’(G)>=(G)>=2.
The graph G-uv is connected
 A u,v-path in G-uv is internally disjoint in G from the u,v-path formed by the edge uv itself.
Induction Step: d(u,v)>1.
1. Let k=d(u,v).
2. Let w be the vertex before v on a shortest u,v-path.
 d(u,w)=k-1.
 G has internally disjoint u,w-paths P and Q.

4. Theorem 4.2.2
3. If vV(P)V(Q), then we find the desired paths in the cycle PQ. Q w u v P

5. Theorem 4.2.2
4. Otherwise, G-w is connected and contains a u,v-path R
since G is 2-connected.
5. If R avoids P or Q, we are done. Q w v P u R

6. Theorem 4.2.2
6. Otherwise, let z be the last vertex of R (before v) belonging to PQ. We assume that zP by symmetry.
 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. R z P w u v Q

7. 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.

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

9. 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.

10. 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. x y e f
 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. u x y z f y x e

11. Theorem 4.2.4
7. For CD. Suppose G satisfies condition C.  G satisfies condition A.  G is connected.  (G)>=1.
8. Consider two edges uv and xy.
Add to G the vertices w with neighborhood {u,v} and
z with neighborhood {x,y} to form G’.
9. Since G is 2-connected, Lemma implies G’ is 2-connected.
 w and z lie on a cycle C in G’. u v x y w z

12. 8. 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.
9. 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

13. 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.

14. 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.

15. 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).

16. Theorem 4.2.17 (Menger Theorem)
Basis Step: n(G)=2.
Induction Step: n(G)>2.
xyE(G) yields (x,y)= (x,y)=0.
1. Let k= G(x,y).
2. N(x) and N(y) are x,y-cuts.
 No minimum cut properly contains N(x) or N(y).
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).

17. Theorem
5. Case 1: G has a minimum x,y-cut S other than N(x) or N(y).
6. Let V1 be the set of vertices on x,S-path, and
let V2 be the set of vertices on S,y-path.
7. SV1 and SV2  SV1V2.
8. 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.  S=V1V2. S V1 V2 x y G v

18. Theorem
9. V1 omits N(y)-S and V2 omits N(x)-S by the same argument in 7.
10. Form H1 by adding to G[V1] a vertex y’ with edges from S.
11. Form H2 by adding to G[V2] a vertex x’ with edges from S. H2 x’ y H1 y’ x

19. Theorem
12. 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.
13. H2(x’,y)= k by the same argument in 8.
14. V1 omits N(y)-S and V2 omits N(x)-S.
 H1 and H2 are smaller than G.
 H1(x,y’)=k= H2(x’,y).

20. Theorem
14. 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. 

21. Theorem 4.2.17 15. Case 2: Every minimum x,y-cut is N(x) or N(y).
16. 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.
 Combining these k-1 x,y-path and the path x,u,y yields k internally disjoint x,y-paths in G.

22. Theorem
17. 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.

23. Theorem
18. We may assume that N(x) and N(y) partition V(G)-{x,y}.
19. Let G’ be the bipartite graph with bipartition N(x), N(y) and edge set [N(x),N(y)].
20. 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
 These k edges yield k internally disjoint x,y-paths of length 3.

24. 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.

25. Theorem
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.
Proof. 1. Modify G to obtain G’ by adding two new vertices s, t and two new edges sx and yt. Cleary, ’G(x,y)= ’G’(x,y) and ’G(x,y)= ’G’(x,y). a d f y x s c t b g e

26. Theorem
2. 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 .

27. Theorem
3. 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).
4. A set of edges disconnects y from x if and only if the corresponding vertices of L(G’) form an sx,yt-cut.  ’G’(x,y)=L(G’)(sx,yt).
5. L(G’)(sx,yt)= L(G’)(sx,yt).  ’G’(x,y)= ’G’(x,y).  ’G(x,y)= ’G(x,y).

28. Theorem
The connectivity of G equals the maximum k such that (x,y)>=k for all x,yV(G).
That is, ’(G)= minx,yV(G) ’(x,y).
The edge-connectivity of G equals the maximum k such that ’(x,y)>=k for all x,yV(G).
That is, (G)=minx,yV(G)(x,y).