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 of if part)
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) G is 2-connected. (Premise)
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 v u w P Q 4.4. If vV(P)V(Q), then
we find the desired paths in the cycle PQ. v u w P Q

6. Theorem 4.2.2 Q w v P u R since G is 2-connected.
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. Q w v P u R

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. R z P w u v Q

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).
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.

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. u x y z f x y e 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 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.
Let k= G(x,y).
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
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. S V1 V2 x y G v

19. Theorem
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. H2 x’ y H1 y’ x

20. Theorem
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
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
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
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
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
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
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.
2. 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

27. Theorem
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. 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).