A graph G is ** k-connected** if G is connected and

The ** edge-connectivity** of a connected graph G, denoted

A graph G is ** k-edge-connected** if G is connected and every edge-cut has at least k edges.

**Proposition 5.1.1 :** Let G be a connected graph. Then the edge-connectivity _{e}(G) is less than or equal to the minimum degree of any vertex of G (_{min}(G)).

**Proposition 5.1.3:** Let e be any edge of a k-connected graph G, for k3. Then the edge-deletion subgraph G-e is (k-1)-connected.

**Corollary 5.1.4:** Let G be a k-connected graph, and let D be any set of m edges of G, for mk-1. Then the edge-deletion subgraph G-D is (k-m)-connected.

**Corollary 5.1.5:** If G is a connected graph, then _{v}(G)_{e}(G).

Thus we always have

**Theorem 5.1.7:** (Whitney, 1932) Let G be a connected graph with at least 3 vertices. G is 2-connected if and only if for each pair of vertices in G, there are two internally disjoint paths between them.

**Corollary 5.1.8:** Let G be a graph with at least three vertices. Then G is 2-connected if and only if any two vertices of G lie on a common cycle.

A ** Whitney synthesis** of a graph G from a graph H is a sequence of graphs starting at H and ending at G, where each graph in the sequence is the result of a path addition to the previous graph.

**Lemma 5.2.1:** Let H be a 2-connected graph. Then the graph that results from a path addition to H is 2-connected.

**Lemma 5.2.2:** Let H be a subgraph of a 2-connected graph G, and let e be any edge in the graph G - E_{H}. Then there is a path addition to H that includes edge e.

**Theorem 5.2.3:** (Whitney, 1932) A graph G is 2-connected if and only if G is a cycle or a Whitney synthesis from a cycle.

**Theorem 5.2.4:** (Tutte, 1961) A graph is 3-connected if and only if it is a wheel or can be obtained from a wheel by a sequence of operations of the following two types.

- Adding an edge between two non-adjacent vertices.
- Replacing a vertex v with degree at least 4 by two new vertices v
^{1}and v^{2}, joined by a new edge; each vertex that was adjacent to v in G is joined by an edge to exactly one of v^{1}and v^{2}so that deg(v^{1}) > 2 and deg(v^{2}) > 2.

**Theorem 5.2.6:** The Harary graph H_{k,n} is k-connected.

**Corollary 5.2.7:** The Harary graph H_{k,n} is a k-edge-connected, n-vertex graph with the fewest possible edges.

**Proposition 5.3.1:** (Weak Duality) Let u and v be any two non-adjacent vertices of a connected graph G. Let P_{uv} be a collection of internally disjoint u-v paths in G, and let S_{uv} be a u-v separating set of vertices in G. Then |P_{uv}||S_{uv}|.

**Corollary 5.3.2:** Let u and v be any two non-adjacent vertices of a connected graph G. Then the maximum number of internally disjoint u-v paths in G is less than or equal to the minimum size of a u-v separating set of vertices of G.

**Theorem 5.3.3:** (Menger, 1927) Let u and v be distinct, non-adjacent vertices in a connected graph G. Then the maximum number of internally disjoint u-v paths in G equals the minimum number of vertices needed to separate u and v.

**Theorem 5.3.6:** (Whitney, 1932) A nontrivial graph G is k-connected if and only if for each pair u, v of vertices, there are at least k internally disjoint u-v paths in G.

**Theorem 5.3.8:** (Dirac, 1960) Let G be a k-connected graph with at least k+1 vertices, for k gt; 2, and let U be any set of k vertices in G. Then there is a cycle in G containing all the vertices in U.