A graph G is k-connected if G is connected and v(G)k. If G has non-adjacent vertices, then G is k-connected if every vertex-cut has at least k vertices.
The edge-connectivity of a connected graph G, denoted e(G), it the minimum number of edges whose removal can disconnect G. Thus, for connected graphs, the edge-connectivity is the size of the smallest edge-cut.
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 - EH. 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.
Theorem 5.2.6: The Harary graph Hk,n is k-connected.
Corollary 5.2.7: The Harary graph Hk,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 Puv be a collection of internally disjoint u-v paths in G, and let Suv be a u-v separating set of vertices in G. Then |Puv||Suv|.
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.