Graph Theory Lecture Notes 8

Vertex and Edge Connectivity

The vertex connectivity of a connected graph G, denoted v(G), is the minimum number of vertices whose removal can either disconnect G or reduce it to a 1-vertex graph. Thus, if G is not a complete graph (i.e., it has at least one pair of non-adjacent vertices), then v(G)is the size of a smallest vertex-cut.

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

v(G)e(G)min(G).

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.

Constructing Reliable Networks

A path addition to a graph G is the addition to G of a path between two existing vertices of G, such that the edges and internal vertices of the path are not in G.

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.

  1. Adding an edge between two non-adjacent vertices.
  2. Replacing a vertex v with degree at least 4 by two new vertices v1 and v2, joined by a new edge; each vertex that was adjacent to v in G is joined by an edge to exactly one of v1 and v2 so that deg(v1) > 2 and deg(v2) > 2.
Harary graphs

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.

Menger's Theorem

Let u and v be distinct vertices in a connected graph G. A vertex subset (or edge subset) S is a u-v separating (or separates u and v), if the vertices u and v lie in different components of the deletion subgraph G - S.

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.