Section 8.4

1. Section 8.4 Connectivity

2. Paths • In an undirected graph, a path of length n from u to v, where n is a positive integer, is a sequence of edges e1, … , en of the graph such that f(e1)={x0,x1}, f(e2)={x1,x2}, … , f(en)={x n-1,xn} where x0 = u and xn = v • In a simple graph, we denote this path by its vertex sequence

3. Paths • Circuit: a path that begins and ends at the same vertex (i.e., u=v) • The path or circuit is said to pass through or traverse the vertices x1, x2, …, xn-1 • Simple path or circuit: path or circuit that does not contain the same edge more than once

4. Example 1 In the simple graph at the left, the path: u1,u2,u4,u5 is a simple path of length 3 since {u1,u2}, {u2,u4}, {u4,u5} all exist as edges

5. Example 1 The path: u1,u2,u5,u4 is not a path, because no edge exists between u2 and u5 The path: u1,u2,u6,u5,u1 is a circuit of length 4

6. Paths in directed multigraphs • Definition is virtually identical to definition of simple graph path • When no multiple edges exist, the graph is denoted by its vertex sequence • A circuit or cycle is a path that begins and ends at the same vertex • A path or circuit is simple if it does not contain the same edge more than once

7. Connectedness in undirected graphs • An undirected graph is connected if there is a path between every pair of distinct vertices • The graph on the right is connected - can find at least one path between every pair of vertices

8. Connected components • A graph that is not connected is the union of two or more connected subgraphs, each pair of which has no vertex in common • These disjoint connected subgraphs are the connected components of the graph

9. Cut vertices and cut edges • A cut vertex (or articulation point) is a vertex which, when removed with all its incident edges, leaves behind a subgraph with more connected components than were found in the original graph • The removal of a cut vertex from a connected graph produces a subgraph that is not connected • An edge whose removal produces a graph with more connected components than in the original graph is called a cut edge or bridge

10. Example Find the cut vertices and cut edges in the graph below:

11. Example Original graph: Vertex c is a cut vertex: Vertex b is a cut vertex: Vertex e is a cut vertex:

12. Example Cut edges are: {a, b} {c, e}

13. Connectedness in digraphs • Strongly connected: a digraph is strongly connected if, for vertices a and b, there is a path from a to b and a pathfrom b to a • Weakly connected: a digraph is weakly connected if there is a path between any two vertices in the underlying undirected graph • A strongly connected graph is also weakly connected, but a weakly connected graph may not be strongly connected

14. Examples Strongly-connected Weakly-connected

15. Paths and Isomorphism • The existence of a simple circuit of length k, where k > 2, is a useful isomorphic invariant for simple graphs • If one graph has such a circuit and the other does not, the graphs are not isomorphic

16. Section 8.4 Connectivity