Connectivity. Definition 1 A separating set (vertex cut) of a connected G is a set S V (G) such that G S has more than one component.

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1 Connectivity Definition 1 A separating set (vertex cut) of a connected G is a set S V (G) such that G S has more than one component. The connectivity, κ(g), is the minimum size of a vertex set S such that G S has more than one component or only one vertex. We say that G is k-connected if κ(g) k.

2 Definition 2 A disconnecting set of edges (edge cut) of a connected G is a set F E(G) such that G F has more than one component. The edge-connectivity, κ (G), is the minimum size of a disconnecting set of edges. We say that G is k-edge-connected if κ (G) k. Theorem 1 (Whitney) For every connected graph G on at least two vertices, κ(g) κ (G) δ(g).

3 Proof. Edges incident to a vertex v with d(v) = δ(g) form an edge cut. Consider a minimum edge cut [S, S] and consider two cases. Case 1: Every vertex of S is adjacent to every vertex of S. Case 2: Let xy / E, x S, y S. Then T := (N(x) S) {w S x : N(w) S } is a separating set.

4 Theorem 2 For every cubic graph G, κ(g) = κ (G). Proof. We have κ(g) κ (G). Let S be a minimum vertex separator. We can assume S 2.

5 v S v S H 1 H 2 H 1 H 2 Lemma 3 (Expansion Lemma) If G is k-connected and G is obtained from G by adding a new vertex with at least k neighbors in G, then G is k-connected.

6 Characterization of 2-connected graphs Definition 3 An ear decomposition of G is a decomposition P 0,..., P k such that P 0 is a cycle and P i is a path with both endpoints in P 0 P i 1 which does not contain any over vertices of P 0 P i 1. Theorem 4 (Whitney) A graph is 2-connected if and only if it has an ear decomposition. Moreover the first cycle of the decomposition can be chosen to be an arbitrary cycle in G. Proof. If there is an ear decomposition, then G is 2-connected: By

7 induction on the number of ears. Show that there are no cut vertices. If G is 2-connected, then there is an ear decomposition: Let C be a cycle (which exists) and let H be a maximal subgraph of G constructible as above.

8 Characterization of 3-connected graphs Let G be simple and let e = {u, v} G. Then G e is the multigraph graph in which we replace u, v with w e and put an edge between x and w e for every edge {x, w} with w {u, v} in G. Let G/e be the simple graph obtained from G e be deleting multiple edges. Lemma 5 (Thomassen) Every 3-connected graph G with V (G) 5 has an edge e = xy such that G/e (and so G e) is 3-connected. Proof.

9 By contradiction. Consider a mate z of {x, y}. Select e = xy so that G {x, y, z} has a component H of the largest order. Show that G[V (H) {x, y}] is 2-connected. Consider {z, u} and its mate v. z u x y H H

10 Theorem 6 A simple graph G is 3-connected if and only if there is a sequence of graphs G 0,..., G n such that 1. G 0 = K 4, G n = G, 2. G i+1 has an edge e = xy such that in G i+1, d(x), d(y) 3 and G i = G i+1 /e.

11 Proof. If G is 3-connected apply the lemma to obtain the sequence G 0,..., G n. Show that if G i is 3-connected then G i+1 is 3-connected. Menger s theorem Let A, B V (G). An A, B-path is a path whose first vertex is in A, last in B, and all internal vertices are neither in A nor in B.

12 Let l(a, B) be the maximum size of a set of disjoint A, B- paths. An A, B-separator is a set S such that there are no A, B-paths in G S. Let k(a, B) be the minimum size of an A, B-separator.

13 Theorem 7 (Menger) Let G be a simple graph and let A, B V (G). Then l(a, B) = k(a, B). Proof. We have l(a, B) k(a, B) as any A, B-separator must contain a vertex from each of the disjoint A, B-paths. k(a, B) l(a, B). Induction on E(G).

14 Let λ(a, b) be the maximum number of internally disjoint a, b- paths. Let κ(a, b) be the minimum size of a set S V (G) \ {a, b} which separates {a} from {b}. Corollary 8 If a, b are non-adjacent then λ(a, b) = κ(a, b). Corollary 9 (Global Version of Menger s theorem) For every simple graph G, κ(g) = t := min a b λ(a, b).

15 Proof. Consider the case of a complete graph. t κ(g). If G is not complete, consider a separator S of size κ(g) and x, y, in distinct components of G S. κ t. If ab / G, follows from the previous corollary. Suppose ab G and there are at most κ 1 internally disjoint a b- paths. Consider G = G ab which has only at most κ 2 internally disjoint a b-paths. Use corollary to separate a from b in G by X with X k 2. Use X to show that G has a separating set of size at most k 1.

16 Definition 4 The line graph H = L(G) of a graph G is the graph with V (H) = E(G) and E(H) = {ef e f }. κ (x, y) - the size of a minimum x, y-edge cut (set of edges F such that there are no x, y-paths in G F). λ (x, y)- the maximum size of a set of edge-disjoint x, y-paths. Theorem 10 Let x, y be two distinct vertices in a graph G. Then κ (x, y) = λ (x, y). Proof. Consider G = G + x + y + xx + yy and the line graph of G.

17 Let x V, U V. An x, U-fan is a set of x, U-paths such that any two paths share only x. Theorem 11 A graph G on n vertices is k-connected if and only if n k + 1 and G has an x, U-fan of sizek for every x V and every U V with U k. If G is k-connected then add a vertex z and make it adjacent to U. Apply the global version of Menger s to x, z. We have δ(g) k and for any x, y there is an x, N(y)-fan of size k.

18 Theorem 12 Let G be a k-connected graph with k 2. Then for any set S V (G) of size k there is a cycle C such that S V (C).

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